Calculus Examples

$y = \frac{9}{5} x^{5} - \frac{47}{3} x^{3} + 10 x$

Step 1

Write $y = \frac{9}{5} x^{5} - \frac{47}{3} x^{3} + 10 x$ as a function.

$f (x) = \frac{9}{5} x^{5} - \frac{47}{3} x^{3} + 10 x$

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $\frac{9}{5} x^{5} - \frac{47}{3} x^{3} + 10 x$ with respect to $x$ is $\frac{d}{d x} [\frac{9}{5} x^{5}] + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$ .

$\frac{d}{d x} [\frac{9}{5} x^{5}] + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{9}{5} x^{5}]$ .

Tap for more steps...

Step 2.2.1

Since $\frac{9}{5}$ is constant with respect to $x$ , the derivative of $\frac{9}{5} x^{5}$ with respect to $x$ is $\frac{9}{5} \frac{d}{d x} [x^{5}]$ .

$\frac{9}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{9}{5} (5 x^{4}) + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.3

Combine $5$ and $\frac{9}{5}$ .

$\frac{5 \cdot 9}{5} x^{4} + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.4

Multiply $5$ by $9$ .

$\frac{45}{5} x^{4} + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.5

Combine $\frac{45}{5}$ and $x^{4}$ .

$\frac{45 x^{4}}{5} + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.6

Cancel the common factor of $45$ and $5$ .

Tap for more steps...

Step 2.2.6.1

Factor $5$ out of $45 x^{4}$ .

$\frac{5 (9 x^{4})}{5} + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.6.2

Cancel the common factors.

Tap for more steps...

Step 2.2.6.2.1

Factor $5$ out of $5$ .

$\frac{5 (9 x^{4})}{5 (1)} + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.6.2.2

Cancel the common factor.

$\frac{5 (9 x^{4})}{5 \cdot 1} + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.6.2.3

Rewrite the expression.

$\frac{9 x^{4}}{1} + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.2.6.2.4

Divide $9 x^{4}$ by $1$ .

$9 x^{4} + \frac{d}{d x} [- \frac{47}{3} x^{3}] + \frac{d}{d x} [10 x]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{47}{3} x^{3}]$ .

Tap for more steps...

Step 2.3.1

Since $- \frac{47}{3}$ is constant with respect to $x$ , the derivative of $- \frac{47}{3} x^{3}$ with respect to $x$ is $- \frac{47}{3} \frac{d}{d x} [x^{3}]$ .

$9 x^{4} - \frac{47}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [10 x]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$9 x^{4} - \frac{47}{3} (3 x^{2}) + \frac{d}{d x} [10 x]$

Step 2.3.3

Multiply $3$ by $- 1$ .

$9 x^{4} - 3 (\frac{47}{3}) x^{2} + \frac{d}{d x} [10 x]$

Step 2.3.4

Combine $- 3$ and $\frac{47}{3}$ .

$9 x^{4} + \frac{- 3 \cdot 47}{3} x^{2} + \frac{d}{d x} [10 x]$

Step 2.3.5

Multiply $- 3$ by $47$ .

$9 x^{4} + \frac{- 141}{3} x^{2} + \frac{d}{d x} [10 x]$

Step 2.3.6

Combine $\frac{- 141}{3}$ and $x^{2}$ .

$9 x^{4} + \frac{- 141 x^{2}}{3} + \frac{d}{d x} [10 x]$

Step 2.3.7

Cancel the common factor of $- 141$ and $3$ .

Tap for more steps...

Step 2.3.7.1

Factor $3$ out of $- 141 x^{2}$ .

$9 x^{4} + \frac{3 (- 47 x^{2})}{3} + \frac{d}{d x} [10 x]$

Step 2.3.7.2

Cancel the common factors.

Tap for more steps...

Step 2.3.7.2.1

Factor $3$ out of $3$ .

$9 x^{4} + \frac{3 (- 47 x^{2})}{3 (1)} + \frac{d}{d x} [10 x]$

Step 2.3.7.2.2

Cancel the common factor.

$9 x^{4} + \frac{3 (- 47 x^{2})}{3 \cdot 1} + \frac{d}{d x} [10 x]$

Step 2.3.7.2.3

Rewrite the expression.

$9 x^{4} + \frac{- 47 x^{2}}{1} + \frac{d}{d x} [10 x]$

Step 2.3.7.2.4

Divide $- 47 x^{2}$ by $1$ .

$9 x^{4} - 47 x^{2} + \frac{d}{d x} [10 x]$

Step 2.4

Evaluate $\frac{d}{d x} [10 x]$ .

Tap for more steps...

Step 2.4.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$9 x^{4} - 47 x^{2} + 10 \frac{d}{d x} [x]$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$9 x^{4} - 47 x^{2} + 10 \cdot 1$

Step 2.4.3

Multiply $10$ by $1$ .

$9 x^{4} - 47 x^{2} + 10$

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $9 x^{4} - 47 x^{2} + 10$ with respect to $x$ is $\frac{d}{d x} [9 x^{4}] + \frac{d}{d x} [- 47 x^{2}] + \frac{d}{d x} [10]$ .

$f'' (x) = \frac{d}{d x} (9 x^{4}) + \frac{d}{d x} (- 47 x^{2}) + \frac{d}{d x} (10)$

Step 3.2

Evaluate $\frac{d}{d x} [9 x^{4}]$ .

Tap for more steps...

Step 3.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{4}$ with respect to $x$ is $9 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = 9 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 47 x^{2}) + \frac{d}{d x} (10)$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = 9 (4 x^{3}) + \frac{d}{d x} (- 47 x^{2}) + \frac{d}{d x} (10)$

Step 3.2.3

Multiply $4$ by $9$ .

$f'' (x) = 36 x^{3} + \frac{d}{d x} (- 47 x^{2}) + \frac{d}{d x} (10)$

Step 3.3

Evaluate $\frac{d}{d x} [- 47 x^{2}]$ .

Tap for more steps...

Step 3.3.1

Since $- 47$ is constant with respect to $x$ , the derivative of $- 47 x^{2}$ with respect to $x$ is $- 47 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 36 x^{3} - 47 \frac{d}{d x} x^{2} + \frac{d}{d x} (10)$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 36 x^{3} - 47 (2 x) + \frac{d}{d x} (10)$

Step 3.3.3

Multiply $2$ by $- 47$ .

$f'' (x) = 36 x^{3} - 94 x + \frac{d}{d x} (10)$

Step 3.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.4.1

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$f'' (x) = 36 x^{3} - 94 x + 0$

Step 3.4.2

Add $36 x^{3} - 94 x$ and $0$ .

$f'' (x) = 36 x^{3} - 94 x$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$9 x^{4} - 47 x^{2} + 10 = 0$

Step 5

Find the first derivative.

Tap for more steps...

Step 5.1

Find the first derivative.

Tap for more steps...

Step 5.1.1

$f' (x) = \frac{d}{d x} (\frac{9}{5} x^{5}) + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2

Evaluate $\frac{d}{d x} [\frac{9}{5} x^{5}]$ .

Tap for more steps...

Step 5.1.2.1

Since $\frac{9}{5}$ is constant with respect to $x$ , the derivative of $\frac{9}{5} x^{5}$ with respect to $x$ is $\frac{9}{5} \frac{d}{d x} [x^{5}]$ .

$f' (x) = \frac{9}{5} \cdot \frac{d}{d x} (x^{5}) + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$f' (x) = \frac{9}{5} \cdot (5 x^{4}) + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.3

Combine $5$ and $\frac{9}{5}$ .

$f' (x) = \frac{5 \cdot 9}{5} x^{4} + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.4

Multiply $5$ by $9$ .

$f' (x) = \frac{45}{5} x^{4} + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.5

Combine $\frac{45}{5}$ and $x^{4}$ .

$f' (x) = \frac{45 x^{4}}{5} + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.6

Cancel the common factor of $45$ and $5$ .

Tap for more steps...

Step 5.1.2.6.1

Factor $5$ out of $45 x^{4}$ .

$f' (x) = \frac{5 (9 x^{4})}{5} + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.6.2

Cancel the common factors.

Tap for more steps...

Step 5.1.2.6.2.1

Factor $5$ out of $5$ .

$f' (x) = \frac{5 (9 x^{4})}{5 (1)} + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.6.2.2

Cancel the common factor.

$f' (x) = \frac{5 (9 x^{4})}{5 \cdot 1} + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.6.2.3

Rewrite the expression.

$f' (x) = \frac{9 x^{4}}{1} + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.2.6.2.4

Divide $9 x^{4}$ by $1$ .

$f' (x) = 9 x^{4} + \frac{d}{d x} (- \frac{47}{3} x^{3}) + \frac{d}{d x} (10 x)$

Step 5.1.3

Evaluate $\frac{d}{d x} [- \frac{47}{3} x^{3}]$ .

Tap for more steps...

Step 5.1.3.1

Since $- \frac{47}{3}$ is constant with respect to $x$ , the derivative of $- \frac{47}{3} x^{3}$ with respect to $x$ is $- \frac{47}{3} \frac{d}{d x} [x^{3}]$ .

$f' (x) = 9 x^{4} - \frac{47}{3} \cdot \frac{d}{d x} x^{3} + \frac{d}{d x} (10 x)$

Step 5.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 9 x^{4} - \frac{47}{3} \cdot (3 x^{2}) + \frac{d}{d x} (10 x)$

Step 5.1.3.3

Multiply $3$ by $- 1$ .

$f' (x) = 9 x^{4} - 3 (\frac{47}{3}) \cdot x^{2} + \frac{d}{d x} (10 x)$

Step 5.1.3.4

Combine $- 3$ and $\frac{47}{3}$ .

$f' (x) = 9 x^{4} + \frac{- 3 \cdot 47}{3} x^{2} + \frac{d}{d x} (10 x)$

Step 5.1.3.5

Multiply $- 3$ by $47$ .

$f' (x) = 9 x^{4} + \frac{- 141}{3} x^{2} + \frac{d}{d x} (10 x)$

Step 5.1.3.6

Combine $\frac{- 141}{3}$ and $x^{2}$ .

$f' (x) = 9 x^{4} + \frac{- 141 x^{2}}{3} + \frac{d}{d x} (10 x)$

Step 5.1.3.7

Cancel the common factor of $- 141$ and $3$ .

Tap for more steps...

Step 5.1.3.7.1

Factor $3$ out of $- 141 x^{2}$ .

$f' (x) = 9 x^{4} + \frac{3 (- 47 x^{2})}{3} + \frac{d}{d x} (10 x)$

Step 5.1.3.7.2

Cancel the common factors.

Tap for more steps...

Step 5.1.3.7.2.1

Factor $3$ out of $3$ .

$f' (x) = 9 x^{4} + \frac{3 (- 47 x^{2})}{3 (1)} + \frac{d}{d x} (10 x)$

Step 5.1.3.7.2.2

Cancel the common factor.

$f' (x) = 9 x^{4} + \frac{3 (- 47 x^{2})}{3 \cdot 1} + \frac{d}{d x} (10 x)$

Step 5.1.3.7.2.3

Rewrite the expression.

$f' (x) = 9 x^{4} + \frac{- 47 x^{2}}{1} + \frac{d}{d x} (10 x)$

Step 5.1.3.7.2.4

Divide $- 47 x^{2}$ by $1$ .

$f' (x) = 9 x^{4} - 47 x^{2} + \frac{d}{d x} (10 x)$

Step 5.1.4

Evaluate $\frac{d}{d x} [10 x]$ .

Tap for more steps...

Step 5.1.4.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$f' (x) = 9 x^{4} - 47 x^{2} + 10 \frac{d}{d x} (x)$

Step 5.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = 9 x^{4} - 47 x^{2} + 10 \cdot 1$

Step 5.1.4.3

Multiply $10$ by $1$ .

$f' (x) = 9 x^{4} - 47 x^{2} + 10$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $9 x^{4} - 47 x^{2} + 10$ .

$9 x^{4} - 47 x^{2} + 10$

Step 6

Set the first derivative equal to $0$ then solve the equation $9 x^{4} - 47 x^{2} + 10 = 0$ .

Tap for more steps...

Step 6.1

Set the first derivative equal to $0$ .

$9 x^{4} - 47 x^{2} + 10 = 0$

Step 6.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$9 u^{2} - 47 u + 10 = 0$

$u = x^{2}$

Step 6.3

Factor by grouping.

Tap for more steps...

Step 6.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot 10 = 90$ and whose sum is $b = - 47$ .

Tap for more steps...

Step 6.3.1.1

Factor $- 47$ out of $- 47 u$ .

$9 u^{2} - 47 u + 10 = 0$

Step 6.3.1.2

Rewrite $- 47$ as $- 2$ plus $- 45$

$9 u^{2} + (- 2 - 45) u + 10 = 0$

Step 6.3.1.3

Apply the distributive property.

$9 u^{2} - 2 u - 45 u + 10 = 0$

Step 6.3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 6.3.2.1

Group the first two terms and the last two terms.

$(9 u^{2} - 2 u) - 45 u + 10 = 0$

Step 6.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (9 u - 2) - 5 (9 u - 2) = 0$

Step 6.3.3

Factor the polynomial by factoring out the greatest common factor, $9 u - 2$ .

$(9 u - 2) (u - 5) = 0$

Step 6.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$9 u - 2 = 0$

$u - 5 = 0$

Step 6.5

Set $9 u - 2$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 6.5.1

Set $9 u - 2$ equal to $0$ .

$9 u - 2 = 0$

Step 6.5.2

Solve $9 u - 2 = 0$ for $u$ .

Tap for more steps...

Step 6.5.2.1

Add $2$ to both sides of the equation.

$9 u = 2$

Step 6.5.2.2

Divide each term in $9 u = 2$ by $9$ and simplify.

Tap for more steps...

Step 6.5.2.2.1

Divide each term in $9 u = 2$ by $9$ .

$\frac{9 u}{9} = \frac{2}{9}$

Step 6.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.5.2.2.2.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 6.5.2.2.2.1.1

Cancel the common factor.

$\frac{9 u}{9} = \frac{2}{9}$

Step 6.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{2}{9}$

Step 6.6

Set $u - 5$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 6.6.1

Set $u - 5$ equal to $0$ .

$u - 5 = 0$

Step 6.6.2

Add $5$ to both sides of the equation.

$u = 5$

Step 6.7

The final solution is all the values that make $(9 u - 2) (u - 5) = 0$ true.

$u = \frac{2}{9}, 5$

Step 6.8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = \frac{2}{9}$

${(x^{2})}^{1} = 5$

Step 6.9

Solve the first equation for $x$ .

$x^{2} = \frac{2}{9}$

Step 6.10

Solve the equation for $x$ .

Tap for more steps...

Step 6.10.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{2}{9}}$

Step 6.10.2

Simplify $\pm \sqrt{\frac{2}{9}}$ .

Tap for more steps...

Step 6.10.2.1

Rewrite $\sqrt{\frac{2}{9}}$ as $\frac{\sqrt{2}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{9}}$

Step 6.10.2.2

Simplify the denominator.

Tap for more steps...

Step 6.10.2.2.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{3^{2}}}$

Step 6.10.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{\sqrt{2}}{3}$

Step 6.10.3

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.10.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{2}}{3}$

Step 6.10.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{2}}{3}$

Step 6.10.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{2}}{3}, - \frac{\sqrt{2}}{3}$

Step 6.11

Solve the second equation for $x$ .

${(x^{2})}^{1} = 5$

Step 6.12

Solve the equation for $x$ .

Tap for more steps...

Step 6.12.1

Remove parentheses.

$x^{2} = 5$

Step 6.12.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{5}$

Step 6.12.3

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.12.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{5}$

Step 6.12.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{5}$

Step 6.12.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{5}, - \sqrt{5}$

Step 6.13

The solution to $9 x^{4} - 47 x^{2} + 10 = 0$ is $x = \frac{\sqrt{2}}{3}, - \frac{\sqrt{2}}{3}, \sqrt{5}, - \sqrt{5}$ .

$x = \frac{\sqrt{2}}{3}, - \frac{\sqrt{2}}{3}, \sqrt{5}, - \sqrt{5}$

Step 7

Find the values where the derivative is undefined.

Tap for more steps...

Step 7.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = \frac{\sqrt{2}}{3}, - \frac{\sqrt{2}}{3}, \sqrt{5}, - \sqrt{5}$

Step 9

Evaluate the second derivative at $x = \frac{\sqrt{2}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$36 {(\frac{\sqrt{2}}{3})}^{3} - 94 \frac{\sqrt{2}}{3}$

Step 10

Evaluate the second derivative.

Tap for more steps...

Step 10.1

Simplify each term.

Tap for more steps...

Step 10.1.1

Apply the product rule to $\frac{\sqrt{2}}{3}$ .

$36 \frac{{\sqrt{2}}^{3}}{3^{3}} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.2

Simplify the numerator.

Tap for more steps...

Step 10.1.2.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$36 \frac{\sqrt{2^{3}}}{3^{3}} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.2.2

Raise $2$ to the power of $3$ .

$36 \frac{\sqrt{8}}{3^{3}} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.2.3

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 10.1.2.3.1

Factor $4$ out of $8$ .

$36 \frac{\sqrt{4 (2)}}{3^{3}} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.2.3.2

Rewrite $4$ as $2^{2}$ .

$36 \frac{\sqrt{2^{2} \cdot 2}}{3^{3}} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.2.4

Pull terms out from under the radical.

$36 \frac{2 \sqrt{2}}{3^{3}} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.3

Raise $3$ to the power of $3$ .

$36 \frac{2 \sqrt{2}}{27} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.4

Cancel the common factor of $9$ .

Tap for more steps...

Step 10.1.4.1

Factor $9$ out of $36$ .

$9 (4) \frac{2 \sqrt{2}}{27} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.4.2

Factor $9$ out of $27$ .

$9 \cdot 4 \frac{2 \sqrt{2}}{9 \cdot 3} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.4.3

Cancel the common factor.

$9 \cdot 4 \frac{2 \sqrt{2}}{9 \cdot 3} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.4.4

Rewrite the expression.

$4 \frac{2 \sqrt{2}}{3} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.5

Combine $4$ and $\frac{2 \sqrt{2}}{3}$ .

$\frac{4 (2 \sqrt{2})}{3} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.6

Multiply $2$ by $4$ .

$\frac{8 \sqrt{2}}{3} - 94 \frac{\sqrt{2}}{3}$

Step 10.1.7

Combine $- 94$ and $\frac{\sqrt{2}}{3}$ .

$\frac{8 \sqrt{2}}{3} + \frac{- 94 \sqrt{2}}{3}$

Step 10.1.8

Move the negative in front of the fraction.

$\frac{8 \sqrt{2}}{3} - \frac{94 \sqrt{2}}{3}$

Step 10.2

Simplify terms.

Tap for more steps...

Step 10.2.1

Combine the numerators over the common denominator.

$\frac{8 \sqrt{2} - 94 \sqrt{2}}{3}$

Step 10.2.2

Subtract $94 \sqrt{2}$ from $8 \sqrt{2}$ .

$\frac{- 86 \sqrt{2}}{3}$

Step 10.2.3

Move the negative in front of the fraction.

$- \frac{86 \sqrt{2}}{3}$

Step 11

$x = \frac{\sqrt{2}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{\sqrt{2}}{3}$ is a local maximum

Step 12

Find the y-value when $x = \frac{\sqrt{2}}{3}$ .

Tap for more steps...

Step 12.1

Replace the variable $x$ with $\frac{\sqrt{2}}{3}$ in the expression.

$f (\frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot {(\frac{\sqrt{2}}{3})}^{5} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2

Simplify the result.

Tap for more steps...

Step 12.2.1

Simplify each term.

Tap for more steps...

Step 12.2.1.1

Apply the product rule to $\frac{\sqrt{2}}{3}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot \frac{{\sqrt{2}}^{5}}{3^{5}} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.2

Combine.

$f (\frac{\sqrt{2}}{3}) = \frac{9 {\sqrt{2}}^{5}}{5 \cdot 3^{5}} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.3

Simplify the numerator.

Tap for more steps...

Step 12.2.1.3.1

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{9 \sqrt{2^{5}}}{5 \cdot 3^{5}} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.3.2

Raise $2$ to the power of $5$ .

$f (\frac{\sqrt{2}}{3}) = \frac{9 \sqrt{32}}{5 \cdot 3^{5}} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.3.3

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 12.2.1.3.3.1

Factor $16$ out of $32$ .

$f (\frac{\sqrt{2}}{3}) = \frac{9 \sqrt{16 (2)}}{5 \cdot 3^{5}} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.3.3.2

Rewrite $16$ as $4^{2}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{9 \sqrt{4^{2} \cdot 2}}{5 \cdot 3^{5}} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.3.4

Pull terms out from under the radical.

$f (\frac{\sqrt{2}}{3}) = \frac{9 \cdot (4 \sqrt{2})}{5 \cdot 3^{5}} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.3.5

Multiply $9$ by $4$ .

$f (\frac{\sqrt{2}}{3}) = \frac{36 \sqrt{2}}{5 \cdot 3^{5}} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.4

Raise $3$ to the power of $5$ .

$f (\frac{\sqrt{2}}{3}) = \frac{36 \sqrt{2}}{5 \cdot 243} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.5

Multiply $5$ by $243$ .

$f (\frac{\sqrt{2}}{3}) = \frac{36 \sqrt{2}}{1215} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.6

Cancel the common factor of $36$ and $1215$ .

Tap for more steps...

Step 12.2.1.6.1

Factor $9$ out of $36 \sqrt{2}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{9 (4 \sqrt{2})}{1215} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.6.2

Cancel the common factors.

Tap for more steps...

Step 12.2.1.6.2.1

Factor $9$ out of $1215$ .

$f (\frac{\sqrt{2}}{3}) = \frac{9 (4 \sqrt{2})}{9 (135)} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.6.2.2

Cancel the common factor.

$f (\frac{\sqrt{2}}{3}) = \frac{9 (4 \sqrt{2})}{9 \cdot 135} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.6.2.3

Rewrite the expression.

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot {(\frac{\sqrt{2}}{3})}^{3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.7

Apply the product rule to $\frac{\sqrt{2}}{3}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot \frac{{\sqrt{2}}^{3}}{3^{3}} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.8

Simplify the numerator.

Tap for more steps...

Step 12.2.1.8.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot \frac{\sqrt{2^{3}}}{3^{3}} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.8.2

Raise $2$ to the power of $3$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot \frac{\sqrt{8}}{3^{3}} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.8.3

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 12.2.1.8.3.1

Factor $4$ out of $8$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot \frac{\sqrt{4 (2)}}{3^{3}} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.8.3.2

Rewrite $4$ as $2^{2}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot \frac{\sqrt{2^{2} \cdot 2}}{3^{3}} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.8.4

Pull terms out from under the radical.

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot \frac{2 \sqrt{2}}{3^{3}} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.9

Raise $3$ to the power of $3$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot \frac{2 \sqrt{2}}{27} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.10

Multiply $- \frac{47}{3} \cdot \frac{2 \sqrt{2}}{27}$ .

Tap for more steps...

Step 12.2.1.10.1

Multiply $\frac{2 \sqrt{2}}{27}$ by $\frac{47}{3}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{2 \sqrt{2} \cdot 47}{27 \cdot 3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.10.2

Multiply $47$ by $2$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{94 \sqrt{2}}{27 \cdot 3} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.10.3

Multiply $27$ by $3$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{94 \sqrt{2}}{81} + 10 (\frac{\sqrt{2}}{3})$

Step 12.2.1.11

Combine $10$ and $\frac{\sqrt{2}}{3}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} - \frac{94 \sqrt{2}}{81} + \frac{10 \sqrt{2}}{3}$

Step 12.2.2

Find the common denominator.

Tap for more steps...

Step 12.2.2.1

Multiply $\frac{4 \sqrt{2}}{135}$ by $\frac{3}{3}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2}}{135} \cdot \frac{3}{3} - \frac{94 \sqrt{2}}{81} + \frac{10 \sqrt{2}}{3}$

Step 12.2.2.2

Multiply $\frac{4 \sqrt{2}}{135}$ by $\frac{3}{3}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} - \frac{94 \sqrt{2}}{81} + \frac{10 \sqrt{2}}{3}$

Step 12.2.2.3

Multiply $\frac{94 \sqrt{2}}{81}$ by $\frac{5}{5}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} - (\frac{94 \sqrt{2}}{81} \cdot \frac{5}{5}) + \frac{10 \sqrt{2}}{3}$

Step 12.2.2.4

Multiply $\frac{94 \sqrt{2}}{81}$ by $\frac{5}{5}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} - \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} + \frac{10 \sqrt{2}}{3}$

Step 12.2.2.5

Multiply $\frac{10 \sqrt{2}}{3}$ by $\frac{135}{135}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} - \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} + \frac{10 \sqrt{2}}{3} \cdot \frac{135}{135}$

Step 12.2.2.6

Multiply $\frac{10 \sqrt{2}}{3}$ by $\frac{135}{135}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} - \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} + \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 12.2.2.7

Reorder the factors of $135 \cdot 3$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{3 \cdot 135} - \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} + \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 12.2.2.8

Multiply $3$ by $135$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{405} - \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} + \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 12.2.2.9

Reorder the factors of $81 \cdot 5$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{405} - \frac{94 \sqrt{2} \cdot 5}{5 \cdot 81} + \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 12.2.2.10

Multiply $5$ by $81$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{405} - \frac{94 \sqrt{2} \cdot 5}{405} + \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 12.2.2.11

Multiply $3$ by $135$ .

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3}{405} - \frac{94 \sqrt{2} \cdot 5}{405} + \frac{10 \sqrt{2} \cdot 135}{405}$

Step 12.2.3

Combine the numerators over the common denominator.

$f (\frac{\sqrt{2}}{3}) = \frac{4 \sqrt{2} \cdot 3 - 94 \sqrt{2} \cdot 5 + 10 \sqrt{2} \cdot 135}{405}$

Step 12.2.4

Simplify each term.

Tap for more steps...

Step 12.2.4.1

Multiply $3$ by $4$ .

$f (\frac{\sqrt{2}}{3}) = \frac{12 \sqrt{2} - 94 \sqrt{2} \cdot 5 + 10 \sqrt{2} \cdot 135}{405}$

Step 12.2.4.2

Multiply $5$ by $- 94$ .

$f (\frac{\sqrt{2}}{3}) = \frac{12 \sqrt{2} - 470 \sqrt{2} + 10 \sqrt{2} \cdot 135}{405}$

Step 12.2.4.3

Multiply $135$ by $10$ .

$f (\frac{\sqrt{2}}{3}) = \frac{12 \sqrt{2} - 470 \sqrt{2} + 1350 \sqrt{2}}{405}$

Step 12.2.5

Simplify by adding terms.

Tap for more steps...

Step 12.2.5.1

Subtract $470 \sqrt{2}$ from $12 \sqrt{2}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{- 458 \sqrt{2} + 1350 \sqrt{2}}{405}$

Step 12.2.5.2

Add $- 458 \sqrt{2}$ and $1350 \sqrt{2}$ .

$f (\frac{\sqrt{2}}{3}) = \frac{892 \sqrt{2}}{405}$

Step 12.2.6

The final answer is $\frac{892 \sqrt{2}}{405}$ .

$y = \frac{892 \sqrt{2}}{405}$

Step 13

Evaluate the second derivative at $x = - \frac{\sqrt{2}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$36 {(- \frac{\sqrt{2}}{3})}^{3} - 94 (- \frac{\sqrt{2}}{3})$

Step 14

Evaluate the second derivative.

Tap for more steps...

Step 14.1

Simplify each term.

Tap for more steps...

Step 14.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 14.1.1.1

Apply the product rule to $- \frac{\sqrt{2}}{3}$ .

$36 ({(- 1)}^{3} {(\frac{\sqrt{2}}{3})}^{3}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.1.2

Apply the product rule to $\frac{\sqrt{2}}{3}$ .

$36 ({(- 1)}^{3} \frac{{\sqrt{2}}^{3}}{3^{3}}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.2

Raise $- 1$ to the power of $3$ .

$36 (- \frac{{\sqrt{2}}^{3}}{3^{3}}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.3

Simplify the numerator.

Tap for more steps...

Step 14.1.3.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$36 (- \frac{\sqrt{2^{3}}}{3^{3}}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.3.2

Raise $2$ to the power of $3$ .

$36 (- \frac{\sqrt{8}}{3^{3}}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.3.3

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 14.1.3.3.1

Factor $4$ out of $8$ .

$36 (- \frac{\sqrt{4 (2)}}{3^{3}}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.3.3.2

Rewrite $4$ as $2^{2}$ .

$36 (- \frac{\sqrt{2^{2} \cdot 2}}{3^{3}}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.3.4

Pull terms out from under the radical.

$36 (- \frac{2 \sqrt{2}}{3^{3}}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.4

Raise $3$ to the power of $3$ .

$36 (- \frac{2 \sqrt{2}}{27}) - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.5

Cancel the common factor of $9$ .

Tap for more steps...

Step 14.1.5.1

Move the leading negative in $- \frac{2 \sqrt{2}}{27}$ into the numerator.

$36 \frac{- 2 \sqrt{2}}{27} - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.5.2

Factor $9$ out of $36$ .

$9 (4) \frac{- 2 \sqrt{2}}{27} - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.5.3

Factor $9$ out of $27$ .

$9 \cdot 4 \frac{- 2 \sqrt{2}}{9 \cdot 3} - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.5.4

Cancel the common factor.

$9 \cdot 4 \frac{- 2 \sqrt{2}}{9 \cdot 3} - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.5.5

Rewrite the expression.

$4 \frac{- 2 \sqrt{2}}{3} - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.6

Combine $4$ and $\frac{- 2 \sqrt{2}}{3}$ .

$\frac{4 (- 2 \sqrt{2})}{3} - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.7

Multiply $- 2$ by $4$ .

$\frac{- 8 \sqrt{2}}{3} - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.8

Move the negative in front of the fraction.

$- \frac{(8) \sqrt{2}}{3} - 94 (- \frac{\sqrt{2}}{3})$

Step 14.1.9

Multiply $- 94 (- \frac{\sqrt{2}}{3})$ .

Tap for more steps...

Step 14.1.9.1

Multiply $- 1$ by $- 94$ .

$- \frac{8 \sqrt{2}}{3} + 94 \frac{\sqrt{2}}{3}$

Step 14.1.9.2

Combine $94$ and $\frac{\sqrt{2}}{3}$ .

$- \frac{8 \sqrt{2}}{3} + \frac{94 \sqrt{2}}{3}$

Step 14.2

Simplify terms.

Tap for more steps...

Step 14.2.1

Combine the numerators over the common denominator.

$\frac{- 8 \sqrt{2} + 94 \sqrt{2}}{3}$

Step 14.2.2

Add $- 8 \sqrt{2}$ and $94 \sqrt{2}$ .

$\frac{86 \sqrt{2}}{3}$

Step 15

$x = - \frac{\sqrt{2}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{\sqrt{2}}{3}$ is a local minimum

Step 16

Find the y-value when $x = - \frac{\sqrt{2}}{3}$ .

Tap for more steps...

Step 16.1

Replace the variable $x$ with $- \frac{\sqrt{2}}{3}$ in the expression.

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot {(- \frac{\sqrt{2}}{3})}^{5} - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2

Simplify the result.

Tap for more steps...

Step 16.2.1

Simplify each term.

Tap for more steps...

Step 16.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 16.2.1.1.1

Apply the product rule to $- \frac{\sqrt{2}}{3}$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot ({(- 1)}^{5} {(\frac{\sqrt{2}}{3})}^{5}) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.1.2

Apply the product rule to $\frac{\sqrt{2}}{3}$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot ({(- 1)}^{5} (\frac{{\sqrt{2}}^{5}}{3^{5}})) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.2

Raise $- 1$ to the power of $5$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot (- \frac{{\sqrt{2}}^{5}}{3^{5}}) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.3

Simplify the numerator.

Tap for more steps...

Step 16.2.1.3.1

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot (- \frac{\sqrt{2^{5}}}{3^{5}}) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.3.2

Raise $2$ to the power of $5$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot (- \frac{\sqrt{32}}{3^{5}}) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.3.3

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 16.2.1.3.3.1

Factor $16$ out of $32$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot (- \frac{\sqrt{16 (2)}}{3^{5}}) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.3.3.2

Rewrite $16$ as $4^{2}$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot (- \frac{\sqrt{4^{2} \cdot 2}}{3^{5}}) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.3.4

Pull terms out from under the radical.

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot (- \frac{4 \sqrt{2}}{3^{5}}) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.4

Raise $3$ to the power of $5$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot (- \frac{4 \sqrt{2}}{243}) - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.5

Cancel the common factor of $9$ .

Tap for more steps...

Step 16.2.1.5.1

Move the leading negative in $- \frac{4 \sqrt{2}}{243}$ into the numerator.

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot \frac{- 4 \sqrt{2}}{243} - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.5.2

Factor $9$ out of $243$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot \frac{- 4 \sqrt{2}}{9 (27)} - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.5.3

Cancel the common factor.

$f (- \frac{\sqrt{2}}{3}) = \frac{9}{5} \cdot \frac{- 4 \sqrt{2}}{9 \cdot 27} - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.5.4

Rewrite the expression.

$f (- \frac{\sqrt{2}}{3}) = \frac{1}{5} \cdot \frac{- 4 \sqrt{2}}{27} - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.6

Multiply $\frac{1}{5}$ by $\frac{- 4 \sqrt{2}}{27}$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{- 4 \sqrt{2}}{5 \cdot 27} - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.7

Multiply $5$ by $27$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{- 4 \sqrt{2}}{135} - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.8

Move the negative in front of the fraction.

$f (- \frac{\sqrt{2}}{3}) = - \frac{(4) \sqrt{2}}{135} - \frac{47}{3} \cdot {(- \frac{\sqrt{2}}{3})}^{3} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 16.2.1.9.1

Apply the product rule to $- \frac{\sqrt{2}}{3}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot ({(- 1)}^{3} {(\frac{\sqrt{2}}{3})}^{3}) + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.9.2

Apply the product rule to $\frac{\sqrt{2}}{3}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} - \frac{47}{3} \cdot ({(- 1)}^{3} (\frac{{\sqrt{2}}^{3}}{3^{3}})) + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.10

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

Tap for more steps...

Step 16.2.1.10.1

Move ${(- 1)}^{3}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{3} \cdot (- 1 (\frac{47}{3})) \cdot \frac{{\sqrt{2}}^{3}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.10.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

Tap for more steps...

Step 16.2.1.10.2.1

Raise $- 1$ to the power of $1$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{3} \cdot ((- 1) (\frac{47}{3})) \cdot \frac{{\sqrt{2}}^{3}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{3 + 1} (\frac{47}{3}) \cdot \frac{{\sqrt{2}}^{3}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.10.3

Add $3$ and $1$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{4} (\frac{47}{3}) \cdot \frac{{\sqrt{2}}^{3}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.11

Simplify the numerator.

Tap for more steps...

Step 16.2.1.11.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{4} (\frac{47}{3}) \cdot \frac{\sqrt{2^{3}}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.11.2

Raise $2$ to the power of $3$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{4} (\frac{47}{3}) \cdot \frac{\sqrt{8}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.11.3

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 16.2.1.11.3.1

Factor $4$ out of $8$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{4} (\frac{47}{3}) \cdot \frac{\sqrt{4 (2)}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.11.3.2

Rewrite $4$ as $2^{2}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{4} (\frac{47}{3}) \cdot \frac{\sqrt{2^{2} \cdot 2}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.11.4

Pull terms out from under the radical.

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{4} (\frac{47}{3}) \cdot \frac{2 \sqrt{2}}{3^{3}} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.12

Raise $3$ to the power of $3$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + {(- 1)}^{4} (\frac{47}{3}) \cdot \frac{2 \sqrt{2}}{27} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.13

Raise $- 1$ to the power of $4$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + 1 (\frac{47}{3}) \cdot \frac{2 \sqrt{2}}{27} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.14

Multiply $\frac{47}{3}$ by $1$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + \frac{47}{3} \cdot \frac{2 \sqrt{2}}{27} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.15

Multiply $\frac{47}{3} \cdot \frac{2 \sqrt{2}}{27}$ .

Tap for more steps...

Step 16.2.1.15.1

Multiply $\frac{47}{3}$ by $\frac{2 \sqrt{2}}{27}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + \frac{47 (2 \sqrt{2})}{3 \cdot 27} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.15.2

Multiply $2$ by $47$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + \frac{94 \sqrt{2}}{3 \cdot 27} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.15.3

Multiply $3$ by $27$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + \frac{94 \sqrt{2}}{81} + 10 (- \frac{\sqrt{2}}{3})$

Step 16.2.1.16

Multiply $10 (- \frac{\sqrt{2}}{3})$ .

Tap for more steps...

Step 16.2.1.16.1

Multiply $- 1$ by $10$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + \frac{94 \sqrt{2}}{81} - 10 \frac{\sqrt{2}}{3}$

Step 16.2.1.16.2

Combine $- 10$ and $\frac{\sqrt{2}}{3}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + \frac{94 \sqrt{2}}{81} + \frac{- 10 \sqrt{2}}{3}$

Step 16.2.1.17

Move the negative in front of the fraction.

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2}}{135} + \frac{94 \sqrt{2}}{81} - \frac{10 \sqrt{2}}{3}$

Step 16.2.2

Find the common denominator.

Tap for more steps...

Step 16.2.2.1

Multiply $\frac{4 \sqrt{2}}{135}$ by $\frac{3}{3}$ .

$f (- \frac{\sqrt{2}}{3}) = - (\frac{4 \sqrt{2}}{135} \cdot \frac{3}{3}) + \frac{94 \sqrt{2}}{81} - \frac{10 \sqrt{2}}{3}$

Step 16.2.2.2

Multiply $\frac{4 \sqrt{2}}{135}$ by $\frac{3}{3}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} + \frac{94 \sqrt{2}}{81} - \frac{10 \sqrt{2}}{3}$

Step 16.2.2.3

Multiply $\frac{94 \sqrt{2}}{81}$ by $\frac{5}{5}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} + \frac{94 \sqrt{2}}{81} \cdot \frac{5}{5} - \frac{10 \sqrt{2}}{3}$

Step 16.2.2.4

Multiply $\frac{94 \sqrt{2}}{81}$ by $\frac{5}{5}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} + \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} - \frac{10 \sqrt{2}}{3}$

Step 16.2.2.5

Multiply $\frac{10 \sqrt{2}}{3}$ by $\frac{135}{135}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} + \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} - (\frac{10 \sqrt{2}}{3} \cdot \frac{135}{135})$

Step 16.2.2.6

Multiply $\frac{10 \sqrt{2}}{3}$ by $\frac{135}{135}$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{135 \cdot 3} + \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} - \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 16.2.2.7

Reorder the factors of $135 \cdot 3$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{3 \cdot 135} + \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} - \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 16.2.2.8

Multiply $3$ by $135$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{405} + \frac{94 \sqrt{2} \cdot 5}{81 \cdot 5} - \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 16.2.2.9

Reorder the factors of $81 \cdot 5$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{405} + \frac{94 \sqrt{2} \cdot 5}{5 \cdot 81} - \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 16.2.2.10

Multiply $5$ by $81$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{405} + \frac{94 \sqrt{2} \cdot 5}{405} - \frac{10 \sqrt{2} \cdot 135}{3 \cdot 135}$

Step 16.2.2.11

Multiply $3$ by $135$ .

$f (- \frac{\sqrt{2}}{3}) = - \frac{4 \sqrt{2} \cdot 3}{405} + \frac{94 \sqrt{2} \cdot 5}{405} - \frac{10 \sqrt{2} \cdot 135}{405}$

Step 16.2.3

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{2}}{3}) = \frac{- 4 \sqrt{2} \cdot 3 + 94 \sqrt{2} \cdot 5 - 10 \sqrt{2} \cdot 135}{405}$

Step 16.2.4

Simplify each term.

Tap for more steps...

Step 16.2.4.1

Multiply $3$ by $- 4$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{- 12 \sqrt{2} + 94 \sqrt{2} \cdot 5 - 10 \sqrt{2} \cdot 135}{405}$

Step 16.2.4.2

Multiply $5$ by $94$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{- 12 \sqrt{2} + 470 \sqrt{2} - 10 \sqrt{2} \cdot 135}{405}$

Step 16.2.4.3

Multiply $135$ by $- 10$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{- 12 \sqrt{2} + 470 \sqrt{2} - 1350 \sqrt{2}}{405}$

Step 16.2.5

Simplify by adding terms.

Tap for more steps...

Step 16.2.5.1

Add $- 12 \sqrt{2}$ and $470 \sqrt{2}$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{458 \sqrt{2} - 1350 \sqrt{2}}{405}$

Step 16.2.5.2

Subtract $1350 \sqrt{2}$ from $458 \sqrt{2}$ .

$f (- \frac{\sqrt{2}}{3}) = \frac{- 892 \sqrt{2}}{405}$

Step 16.2.5.3

Move the negative in front of the fraction.

$f (- \frac{\sqrt{2}}{3}) = - \frac{892 \sqrt{2}}{405}$

Step 16.2.6

The final answer is $- \frac{892 \sqrt{2}}{405}$ .

$y = - \frac{892 \sqrt{2}}{405}$

Step 17

Evaluate the second derivative at $x = \sqrt{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$36 {(\sqrt{5})}^{3} - 94 \sqrt{5}$

Step 18

Evaluate the second derivative.

Tap for more steps...

Step 18.1

Simplify each term.

Tap for more steps...

Step 18.1.1

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$36 \sqrt{5^{3}} - 94 \sqrt{5}$

Step 18.1.2

Raise $5$ to the power of $3$ .

$36 \sqrt{125} - 94 \sqrt{5}$

Step 18.1.3

Rewrite $125$ as $5^{2} \cdot 5$ .

Tap for more steps...

Step 18.1.3.1

Factor $25$ out of $125$ .

$36 \sqrt{25 (5)} - 94 \sqrt{5}$

Step 18.1.3.2

Rewrite $25$ as $5^{2}$ .

$36 \sqrt{5^{2} \cdot 5} - 94 \sqrt{5}$

Step 18.1.4

Pull terms out from under the radical.

$36 (5 \sqrt{5}) - 94 \sqrt{5}$

Step 18.1.5

Multiply $5$ by $36$ .

$180 \sqrt{5} - 94 \sqrt{5}$

Step 18.2

Subtract $94 \sqrt{5}$ from $180 \sqrt{5}$ .

$86 \sqrt{5}$

Step 19

$x = \sqrt{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \sqrt{5}$ is a local minimum

Step 20

Find the y-value when $x = \sqrt{5}$ .

Tap for more steps...

Step 20.1

Replace the variable $x$ with $\sqrt{5}$ in the expression.

$f (\sqrt{5}) = \frac{9}{5} \cdot {(\sqrt{5})}^{5} - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2

Simplify the result.

Tap for more steps...

Step 20.2.1

Simplify each term.

Tap for more steps...

Step 20.2.1.1

Rewrite ${\sqrt{5}}^{5}$ as $\sqrt{5^{5}}$ .

$f (\sqrt{5}) = \frac{9}{5} \cdot \sqrt{5^{5}} - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.2

Raise $5$ to the power of $5$ .

$f (\sqrt{5}) = \frac{9}{5} \cdot \sqrt{3125} - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.3

Rewrite $3125$ as $25^{2} \cdot 5$ .

Tap for more steps...

Step 20.2.1.3.1

Factor $625$ out of $3125$ .

$f (\sqrt{5}) = \frac{9}{5} \cdot \sqrt{625 (5)} - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.3.2

Rewrite $625$ as $25^{2}$ .

$f (\sqrt{5}) = \frac{9}{5} \cdot \sqrt{25^{2} \cdot 5} - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.4

Pull terms out from under the radical.

$f (\sqrt{5}) = \frac{9}{5} \cdot (25 \sqrt{5}) - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.5

Cancel the common factor of $5$ .

Tap for more steps...

Step 20.2.1.5.1

Factor $5$ out of $25 \sqrt{5}$ .

$f (\sqrt{5}) = \frac{9}{5} \cdot (5 (5 \sqrt{5})) - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.5.2

Cancel the common factor.

$f (\sqrt{5}) = \frac{9}{5} \cdot (5 (5 \sqrt{5})) - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.5.3

Rewrite the expression.

$f (\sqrt{5}) = 9 \cdot (5 \sqrt{5}) - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.6

Multiply $5$ by $9$ .

$f (\sqrt{5}) = 45 \cdot \sqrt{5} - \frac{47}{3} \cdot {(\sqrt{5})}^{3} + 10 (\sqrt{5})$

Step 20.2.1.7

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (\sqrt{5}) = 45 \sqrt{5} - \frac{47}{3} \cdot \sqrt{5^{3}} + 10 (\sqrt{5})$

Step 20.2.1.8

Raise $5$ to the power of $3$ .

$f (\sqrt{5}) = 45 \sqrt{5} - \frac{47}{3} \cdot \sqrt{125} + 10 (\sqrt{5})$

Step 20.2.1.9

Rewrite $125$ as $5^{2} \cdot 5$ .

Tap for more steps...

Step 20.2.1.9.1

Factor $25$ out of $125$ .

$f (\sqrt{5}) = 45 \sqrt{5} - \frac{47}{3} \cdot \sqrt{25 (5)} + 10 (\sqrt{5})$

Step 20.2.1.9.2

Rewrite $25$ as $5^{2}$ .

$f (\sqrt{5}) = 45 \sqrt{5} - \frac{47}{3} \cdot \sqrt{5^{2} \cdot 5} + 10 (\sqrt{5})$

Step 20.2.1.10

Pull terms out from under the radical.

$f (\sqrt{5}) = 45 \sqrt{5} - \frac{47}{3} \cdot (5 \sqrt{5}) + 10 (\sqrt{5})$

Step 20.2.1.11

Multiply $- \frac{47}{3} (5 \sqrt{5})$ .

Tap for more steps...

Step 20.2.1.11.1

Multiply $5$ by $- 1$ .

$f (\sqrt{5}) = 45 \sqrt{5} - 5 (\frac{47}{3}) \cdot \sqrt{5} + 10 (\sqrt{5})$

Step 20.2.1.11.2

Combine $- 5$ and $\frac{47}{3}$ .

$f (\sqrt{5}) = 45 \sqrt{5} + \frac{- 5 \cdot 47}{3} \cdot \sqrt{5} + 10 (\sqrt{5})$

Step 20.2.1.11.3

Multiply $- 5$ by $47$ .

$f (\sqrt{5}) = 45 \sqrt{5} + \frac{- 235}{3} \cdot \sqrt{5} + 10 (\sqrt{5})$

Step 20.2.1.11.4

Combine $\frac{- 235}{3}$ and $\sqrt{5}$ .

$f (\sqrt{5}) = 45 \sqrt{5} + \frac{- 235 \sqrt{5}}{3} + 10 (\sqrt{5})$

Step 20.2.1.12

Move the negative in front of the fraction.

$f (\sqrt{5}) = 45 \sqrt{5} - \frac{235 \sqrt{5}}{3} + 10 \sqrt{5}$

Step 20.2.2

Find the common denominator.

Tap for more steps...

Step 20.2.2.1

Write $45 \sqrt{5}$ as a fraction with denominator $1$ .

$f (\sqrt{5}) = \frac{45 \sqrt{5}}{1} - \frac{235 \sqrt{5}}{3} + 10 \sqrt{5}$

Step 20.2.2.2

Multiply $\frac{45 \sqrt{5}}{1}$ by $\frac{3}{3}$ .

$f (\sqrt{5}) = \frac{45 \sqrt{5}}{1} \cdot \frac{3}{3} - \frac{235 \sqrt{5}}{3} + 10 \sqrt{5}$

Step 20.2.2.3

Multiply $\frac{45 \sqrt{5}}{1}$ by $\frac{3}{3}$ .

$f (\sqrt{5}) = \frac{45 \sqrt{5} \cdot 3}{3} - \frac{235 \sqrt{5}}{3} + 10 \sqrt{5}$

Step 20.2.2.4

Write $10 \sqrt{5}$ as a fraction with denominator $1$ .

$f (\sqrt{5}) = \frac{45 \sqrt{5} \cdot 3}{3} - \frac{235 \sqrt{5}}{3} + \frac{10 \sqrt{5}}{1}$

Step 20.2.2.5

Multiply $\frac{10 \sqrt{5}}{1}$ by $\frac{3}{3}$ .

$f (\sqrt{5}) = \frac{45 \sqrt{5} \cdot 3}{3} - \frac{235 \sqrt{5}}{3} + \frac{10 \sqrt{5}}{1} \cdot \frac{3}{3}$

Step 20.2.2.6

Multiply $\frac{10 \sqrt{5}}{1}$ by $\frac{3}{3}$ .

$f (\sqrt{5}) = \frac{45 \sqrt{5} \cdot 3}{3} - \frac{235 \sqrt{5}}{3} + \frac{10 \sqrt{5} \cdot 3}{3}$

Step 20.2.3

Combine the numerators over the common denominator.

$f (\sqrt{5}) = \frac{45 \sqrt{5} \cdot 3 - 235 \sqrt{5} + 10 \sqrt{5} \cdot 3}{3}$

Step 20.2.4

Simplify each term.

Tap for more steps...

Step 20.2.4.1

Multiply $3$ by $45$ .

$f (\sqrt{5}) = \frac{135 \sqrt{5} - 235 \sqrt{5} + 10 \sqrt{5} \cdot 3}{3}$

Step 20.2.4.2

Multiply $3$ by $10$ .

$f (\sqrt{5}) = \frac{135 \sqrt{5} - 235 \sqrt{5} + 30 \sqrt{5}}{3}$

Step 20.2.5

Simplify by adding terms.

Tap for more steps...

Step 20.2.5.1

Subtract $235 \sqrt{5}$ from $135 \sqrt{5}$ .

$f (\sqrt{5}) = \frac{- 100 \sqrt{5} + 30 \sqrt{5}}{3}$

Step 20.2.5.2

Add $- 100 \sqrt{5}$ and $30 \sqrt{5}$ .

$f (\sqrt{5}) = \frac{- 70 \sqrt{5}}{3}$

Step 20.2.5.3

Move the negative in front of the fraction.

$f (\sqrt{5}) = - \frac{70 \sqrt{5}}{3}$

Step 20.2.6

The final answer is $- \frac{70 \sqrt{5}}{3}$ .

$y = - \frac{70 \sqrt{5}}{3}$

Step 21

Evaluate the second derivative at $x = - \sqrt{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$36 {(- \sqrt{5})}^{3} - 94 (- \sqrt{5})$

Step 22

Evaluate the second derivative.

Tap for more steps...

Step 22.1

Simplify each term.

Tap for more steps...

Step 22.1.1

Apply the product rule to $- \sqrt{5}$ .

$36 ({(- 1)}^{3} {\sqrt{5}}^{3}) - 94 (- \sqrt{5})$

Step 22.1.2

Raise $- 1$ to the power of $3$ .

$36 (- {\sqrt{5}}^{3}) - 94 (- \sqrt{5})$

Step 22.1.3

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$36 (- \sqrt{5^{3}}) - 94 (- \sqrt{5})$

Step 22.1.4

Raise $5$ to the power of $3$ .

$36 (- \sqrt{125}) - 94 (- \sqrt{5})$

Step 22.1.5

Rewrite $125$ as $5^{2} \cdot 5$ .

Tap for more steps...

Step 22.1.5.1

Factor $25$ out of $125$ .

$36 (- \sqrt{25 (5)}) - 94 (- \sqrt{5})$

Step 22.1.5.2

Rewrite $25$ as $5^{2}$ .

$36 (- \sqrt{5^{2} \cdot 5}) - 94 (- \sqrt{5})$

Step 22.1.6

Pull terms out from under the radical.

$36 (- (5 \sqrt{5})) - 94 (- \sqrt{5})$

Step 22.1.7

Multiply $5$ by $- 1$ .

$36 (- 5 \sqrt{5}) - 94 (- \sqrt{5})$

Step 22.1.8

Multiply $- 5$ by $36$ .

$- 180 \sqrt{5} - 94 (- \sqrt{5})$

Step 22.1.9

Multiply $- 1$ by $- 94$ .

$- 180 \sqrt{5} + 94 \sqrt{5}$

Step 22.2

Add $- 180 \sqrt{5}$ and $94 \sqrt{5}$ .

$- 86 \sqrt{5}$

Step 23

$x = - \sqrt{5}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \sqrt{5}$ is a local maximum

Step 24

Find the y-value when $x = - \sqrt{5}$ .

Tap for more steps...

Step 24.1

Replace the variable $x$ with $- \sqrt{5}$ in the expression.

$f (- \sqrt{5}) = \frac{9}{5} \cdot {(- \sqrt{5})}^{5} - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2

Simplify the result.

Tap for more steps...

Step 24.2.1

Simplify each term.

Tap for more steps...

Step 24.2.1.1

Apply the product rule to $- \sqrt{5}$ .

$f (- \sqrt{5}) = \frac{9}{5} \cdot ({(- 1)}^{5} {\sqrt{5}}^{5}) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.2

Raise $- 1$ to the power of $5$ .

$f (- \sqrt{5}) = \frac{9}{5} \cdot (- {\sqrt{5}}^{5}) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.3

Rewrite ${\sqrt{5}}^{5}$ as $\sqrt{5^{5}}$ .

$f (- \sqrt{5}) = \frac{9}{5} \cdot (- \sqrt{5^{5}}) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.4

Raise $5$ to the power of $5$ .

$f (- \sqrt{5}) = \frac{9}{5} \cdot (- \sqrt{3125}) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.5

Rewrite $3125$ as $25^{2} \cdot 5$ .

Tap for more steps...

Step 24.2.1.5.1

Factor $625$ out of $3125$ .

$f (- \sqrt{5}) = \frac{9}{5} \cdot (- \sqrt{625 (5)}) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.5.2

Rewrite $625$ as $25^{2}$ .

$f (- \sqrt{5}) = \frac{9}{5} \cdot (- \sqrt{25^{2} \cdot 5}) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.6

Pull terms out from under the radical.

$f (- \sqrt{5}) = \frac{9}{5} \cdot (- (25 \sqrt{5})) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.7

Multiply $25$ by $- 1$ .

$f (- \sqrt{5}) = \frac{9}{5} \cdot (- 25 \sqrt{5}) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.8

Cancel the common factor of $5$ .

Tap for more steps...

Step 24.2.1.8.1

Factor $5$ out of $- 25 \sqrt{5}$ .

$f (- \sqrt{5}) = \frac{9}{5} \cdot (5 (- 5 \sqrt{5})) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.8.2

Cancel the common factor.

$f (- \sqrt{5}) = \frac{9}{5} \cdot (5 (- 5 \sqrt{5})) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.8.3

Rewrite the expression.

$f (- \sqrt{5}) = 9 \cdot (- 5 \sqrt{5}) - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.9

Multiply $- 5$ by $9$ .

$f (- \sqrt{5}) = - 45 \cdot \sqrt{5} - \frac{47}{3} \cdot {(- \sqrt{5})}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.10

Apply the product rule to $- \sqrt{5}$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} - \frac{47}{3} \cdot ({(- 1)}^{3} {\sqrt{5}}^{3}) + 10 (- \sqrt{5})$

Step 24.2.1.11

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

Tap for more steps...

Step 24.2.1.11.1

Move ${(- 1)}^{3}$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + {(- 1)}^{3} \cdot (- 1 (\frac{47}{3})) \cdot {\sqrt{5}}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.11.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

Tap for more steps...

Step 24.2.1.11.2.1

Raise $- 1$ to the power of $1$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + {(- 1)}^{3} \cdot ((- 1) (\frac{47}{3})) \cdot {\sqrt{5}}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \sqrt{5}) = - 45 \sqrt{5} + {(- 1)}^{3 + 1} (\frac{47}{3}) \cdot {\sqrt{5}}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.11.3

Add $3$ and $1$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + {(- 1)}^{4} (\frac{47}{3}) \cdot {\sqrt{5}}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.12

Raise $- 1$ to the power of $4$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + 1 (\frac{47}{3}) \cdot {\sqrt{5}}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.13

Multiply $\frac{47}{3}$ by $1$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{47}{3} \cdot {\sqrt{5}}^{3} + 10 (- \sqrt{5})$

Step 24.2.1.14

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{47}{3} \cdot \sqrt{5^{3}} + 10 (- \sqrt{5})$

Step 24.2.1.15

Raise $5$ to the power of $3$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{47}{3} \cdot \sqrt{125} + 10 (- \sqrt{5})$

Step 24.2.1.16

Rewrite $125$ as $5^{2} \cdot 5$ .

Tap for more steps...

Step 24.2.1.16.1

Factor $25$ out of $125$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{47}{3} \cdot \sqrt{25 (5)} + 10 (- \sqrt{5})$

Step 24.2.1.16.2

Rewrite $25$ as $5^{2}$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{47}{3} \cdot \sqrt{5^{2} \cdot 5} + 10 (- \sqrt{5})$

Step 24.2.1.17

Pull terms out from under the radical.

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{47}{3} \cdot (5 \sqrt{5}) + 10 (- \sqrt{5})$

Step 24.2.1.18

Multiply $\frac{47}{3} (5 \sqrt{5})$ .

Tap for more steps...

Step 24.2.1.18.1

Combine $5$ and $\frac{47}{3}$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{5 \cdot 47}{3} \cdot \sqrt{5} + 10 (- \sqrt{5})$

Step 24.2.1.18.2

Multiply $5$ by $47$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{235}{3} \cdot \sqrt{5} + 10 (- \sqrt{5})$

Step 24.2.1.18.3

Combine $\frac{235}{3}$ and $\sqrt{5}$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{235 \sqrt{5}}{3} + 10 (- \sqrt{5})$

Step 24.2.1.19

Multiply $- 1$ by $10$ .

$f (- \sqrt{5}) = - 45 \sqrt{5} + \frac{235 \sqrt{5}}{3} - 10 \sqrt{5}$

Step 24.2.2

Find the common denominator.

Tap for more steps...

Step 24.2.2.1

Write $- 45 \sqrt{5}$ as a fraction with denominator $1$ .

$f (- \sqrt{5}) = \frac{- 45 \sqrt{5}}{1} + \frac{235 \sqrt{5}}{3} - 10 \sqrt{5}$

Step 24.2.2.2

Multiply $\frac{- 45 \sqrt{5}}{1}$ by $\frac{3}{3}$ .

$f (- \sqrt{5}) = \frac{- 45 \sqrt{5}}{1} \cdot \frac{3}{3} + \frac{235 \sqrt{5}}{3} - 10 \sqrt{5}$

Step 24.2.2.3

Multiply $\frac{- 45 \sqrt{5}}{1}$ by $\frac{3}{3}$ .

$f (- \sqrt{5}) = \frac{- 45 \sqrt{5} \cdot 3}{3} + \frac{235 \sqrt{5}}{3} - 10 \sqrt{5}$

Step 24.2.2.4

Write $- 10 \sqrt{5}$ as a fraction with denominator $1$ .

$f (- \sqrt{5}) = \frac{- 45 \sqrt{5} \cdot 3}{3} + \frac{235 \sqrt{5}}{3} + \frac{- 10 \sqrt{5}}{1}$

Step 24.2.2.5

Multiply $\frac{- 10 \sqrt{5}}{1}$ by $\frac{3}{3}$ .

$f (- \sqrt{5}) = \frac{- 45 \sqrt{5} \cdot 3}{3} + \frac{235 \sqrt{5}}{3} + \frac{- 10 \sqrt{5}}{1} \cdot \frac{3}{3}$

Step 24.2.2.6

Multiply $\frac{- 10 \sqrt{5}}{1}$ by $\frac{3}{3}$ .

$f (- \sqrt{5}) = \frac{- 45 \sqrt{5} \cdot 3}{3} + \frac{235 \sqrt{5}}{3} + \frac{- 10 \sqrt{5} \cdot 3}{3}$

Step 24.2.3

Combine the numerators over the common denominator.

$f (- \sqrt{5}) = \frac{- 45 \sqrt{5} \cdot 3 + 235 \sqrt{5} - 10 \sqrt{5} \cdot 3}{3}$

Step 24.2.4

Simplify each term.

Tap for more steps...

Step 24.2.4.1

Multiply $3$ by $- 45$ .

$f (- \sqrt{5}) = \frac{- 135 \sqrt{5} + 235 \sqrt{5} - 10 \sqrt{5} \cdot 3}{3}$

Step 24.2.4.2

Multiply $3$ by $- 10$ .

$f (- \sqrt{5}) = \frac{- 135 \sqrt{5} + 235 \sqrt{5} - 30 \sqrt{5}}{3}$

Step 24.2.5

Simplify by adding terms.

Tap for more steps...

Step 24.2.5.1

Add $- 135 \sqrt{5}$ and $235 \sqrt{5}$ .

$f (- \sqrt{5}) = \frac{100 \sqrt{5} - 30 \sqrt{5}}{3}$

Step 24.2.5.2

Subtract $30 \sqrt{5}$ from $100 \sqrt{5}$ .

$f (- \sqrt{5}) = \frac{70 \sqrt{5}}{3}$

Step 24.2.6

The final answer is $\frac{70 \sqrt{5}}{3}$ .

$y = \frac{70 \sqrt{5}}{3}$

Step 25

These are the local extrema for $f (x) = \frac{9}{5} x^{5} - \frac{47}{3} x^{3} + 10 x$ .

$(\frac{\sqrt{2}}{3}, \frac{892 \sqrt{2}}{405})$ is a local maxima

$(- \frac{\sqrt{2}}{3}, - \frac{892 \sqrt{2}}{405})$ is a local minima

$(\sqrt{5}, - \frac{70 \sqrt{5}}{3})$ is a local minima

$(- \sqrt{5}, \frac{70 \sqrt{5}}{3})$ is a local maxima

Step 26