Calculus Examples

$\frac{x^{5}}{10} + \frac{x^{4}}{8} + 2$

Step 1

Write $\frac{x^{5}}{10} + \frac{x^{4}}{8} + 2$ as a function.

$f (x) = \frac{x^{5}}{10} + \frac{x^{4}}{8} + 2$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $\frac{x^{5}}{10} + \frac{x^{4}}{8} + 2$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{5}}{10}] + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [\frac{x^{5}}{10}] + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{x^{5}}{10}]$ .

Tap for more steps...

Step 2.1.2.1

Since $\frac{1}{10}$ is constant with respect to $x$ , the derivative of $\frac{x^{5}}{10}$ with respect to $x$ is $\frac{1}{10} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{10} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.2.2

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

$\frac{1}{10} (5 x^{4}) + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.2.3

Combine $5$ and $\frac{1}{10}$ .

$\frac{5}{10} x^{4} + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.2.4

Combine $\frac{5}{10}$ and $x^{4}$ .

$\frac{5 x^{4}}{10} + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.2.5

Cancel the common factor of $5$ and $10$ .

Tap for more steps...

Step 2.1.2.5.1

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

$\frac{5 (x^{4})}{10} + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.2.5.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.5.2.1

Factor $5$ out of $10$ .

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.2.5.2.2

Cancel the common factor.

$\frac{5 x^{4}}{5 \cdot 2} + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.2.5.2.3

Rewrite the expression.

$\frac{x^{4}}{2} + \frac{d}{d x} [\frac{x^{4}}{8}] + \frac{d}{d x} [2]$

Step 2.1.3

Evaluate $\frac{d}{d x} [\frac{x^{4}}{8}]$ .

Tap for more steps...

Step 2.1.3.1

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{8}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [x^{4}]$ .

$\frac{x^{4}}{2} + \frac{1}{8} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [2]$

Step 2.1.3.2

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

$\frac{x^{4}}{2} + \frac{1}{8} (4 x^{3}) + \frac{d}{d x} [2]$

Step 2.1.3.3

Combine $4$ and $\frac{1}{8}$ .

$\frac{x^{4}}{2} + \frac{4}{8} x^{3} + \frac{d}{d x} [2]$

Step 2.1.3.4

Combine $\frac{4}{8}$ and $x^{3}$ .

$\frac{x^{4}}{2} + \frac{4 x^{3}}{8} + \frac{d}{d x} [2]$

Step 2.1.3.5

Cancel the common factor of $4$ and $8$ .

Tap for more steps...

Step 2.1.3.5.1

Factor $4$ out of $4 x^{3}$ .

$\frac{x^{4}}{2} + \frac{4 (x^{3})}{8} + \frac{d}{d x} [2]$

Step 2.1.3.5.2

Cancel the common factors.

Tap for more steps...

Step 2.1.3.5.2.1

Factor $4$ out of $8$ .

$\frac{x^{4}}{2} + \frac{4 x^{3}}{4 \cdot 2} + \frac{d}{d x} [2]$

Step 2.1.3.5.2.2

Cancel the common factor.

$\frac{x^{4}}{2} + \frac{4 x^{3}}{4 \cdot 2} + \frac{d}{d x} [2]$

Step 2.1.3.5.2.3

Rewrite the expression.

$\frac{x^{4}}{2} + \frac{x^{3}}{2} + \frac{d}{d x} [2]$

Step 2.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{x^{4}}{2} + \frac{x^{3}}{2} + 0$

Step 2.1.4.2

Add $\frac{x^{4}}{2} + \frac{x^{3}}{2}$ and $0$ .

$f' (x) = \frac{x^{4}}{2} + \frac{x^{3}}{2}$

Step 2.2

Find the second derivative.

Tap for more steps...

Step 2.2.1

By the Sum Rule, the derivative of $\frac{x^{4}}{2} + \frac{x^{3}}{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{4}}{2}] + \frac{d}{d x} [\frac{x^{3}}{2}]$ .

$\frac{d}{d x} [\frac{x^{4}}{2}] + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.2

Evaluate $\frac{d}{d x} [\frac{x^{4}}{2}]$ .

Tap for more steps...

Step 2.2.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.2.2

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

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

Step 2.2.2.3

Combine $4$ and $\frac{1}{2}$ .

$\frac{4}{2} x^{3} + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.2.4

Combine $\frac{4}{2}$ and $x^{3}$ .

$\frac{4 x^{3}}{2} + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.2.5

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 2.2.2.5.1

Factor $2$ out of $4 x^{3}$ .

$\frac{2 (2 x^{3})}{2} + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.2.5.2

Cancel the common factors.

Tap for more steps...

Step 2.2.2.5.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 x^{3})}{2 (1)} + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.2.5.2.2

Cancel the common factor.

$\frac{2 (2 x^{3})}{2 \cdot 1} + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.2.5.2.3

Rewrite the expression.

$\frac{2 x^{3}}{1} + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.2.5.2.4

Divide $2 x^{3}$ by $1$ .

$2 x^{3} + \frac{d}{d x} [\frac{x^{3}}{2}]$

Step 2.2.3

Evaluate $\frac{d}{d x} [\frac{x^{3}}{2}]$ .

Tap for more steps...

Step 2.2.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{3}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{3}]$ .

$2 x^{3} + \frac{1}{2} \frac{d}{d x} [x^{3}]$

Step 2.2.3.2

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

$2 x^{3} + \frac{1}{2} (3 x^{2})$

Step 2.2.3.3

Combine $3$ and $\frac{1}{2}$ .

$2 x^{3} + \frac{3}{2} x^{2}$

Step 2.2.3.4

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

$f'' (x) = 2 x^{3} + \frac{3 x^{2}}{2}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $2 x^{3} + \frac{3 x^{2}}{2}$ .

$2 x^{3} + \frac{3 x^{2}}{2}$

Step 3

Set the second derivative equal to $0$ then solve the equation $2 x^{3} + \frac{3 x^{2}}{2} = 0$ .

Tap for more steps...

Step 3.1

Set the second derivative equal to $0$ .

$2 x^{3} + \frac{3 x^{2}}{2} = 0$

Step 3.2

Factor $x^{2}$ out of $2 x^{3} + \frac{3 x^{2}}{2}$ .

Tap for more steps...

Step 3.2.1

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

$x^{2} (2 x) + \frac{3 x^{2}}{2} = 0$

Step 3.2.2

Factor $x^{2}$ out of $\frac{3 x^{2}}{2}$ .

$x^{2} (2 x) + x^{2} (\frac{3}{2}) = 0$

Step 3.2.3

Factor $x^{2}$ out of $x^{2} (2 x) + x^{2} \frac{3}{2}$ .

$x^{2} (2 x + \frac{3}{2}) = 0$

Step 3.3

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

$x^{2} = 0$

$2 x + \frac{3}{2} = 0$

Step 3.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 3.4.2

Solve $x^{2} = 0$ for $x$ .

Tap for more steps...

Step 3.4.2.1

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

$x = \pm \sqrt{0}$

Step 3.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 3.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.4.2.2.2

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

$x = \pm 0$

Step 3.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.5

Set $2 x + \frac{3}{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.5.1

Set $2 x + \frac{3}{2}$ equal to $0$ .

$2 x + \frac{3}{2} = 0$

Step 3.5.2

Solve $2 x + \frac{3}{2} = 0$ for $x$ .

Tap for more steps...

Step 3.5.2.1

Subtract $\frac{3}{2}$ from both sides of the equation.

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

Step 3.5.2.2

Divide each term in $2 x = - \frac{3}{2}$ by $2$ and simplify.

Tap for more steps...

Step 3.5.2.2.1

Divide each term in $2 x = - \frac{3}{2}$ by $2$ .

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

Step 3.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.2.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.5.2.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{3}{2} \cdot \frac{1}{2}$

Step 3.5.2.2.3.2

Multiply $- \frac{3}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 3.5.2.2.3.2.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

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

Step 3.5.2.2.3.2.2

Multiply $2$ by $2$ .

$x = - \frac{3}{4}$

Step 3.6

The final solution is all the values that make $x^{2} (2 x + \frac{3}{2}) = 0$ true.

$x = 0, - \frac{3}{4}$

Step 4

Find the points where the second derivative is $0$ .

Tap for more steps...

Step 4.1

Substitute $0$ in $f (x) = \frac{x^{5}}{10} + \frac{x^{4}}{8} + 2$ to find the value of $y$ .

Tap for more steps...

Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{{(0)}^{5}}{10} + \frac{{(0)}^{4}}{8} + 2$

Step 4.1.2

Simplify the result.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{0}{10} + \frac{{(0)}^{4}}{8} + 2$

Step 4.1.2.1.2

Divide $0$ by $10$ .

$f (0) = 0 + \frac{{(0)}^{4}}{8} + 2$

Step 4.1.2.1.3

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + \frac{0}{8} + 2$

Step 4.1.2.1.4

Divide $0$ by $8$ .

$f (0) = 0 + 0 + 2$

Step 4.1.2.2

Simplify by adding numbers.

Tap for more steps...

Step 4.1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 2$

Step 4.1.2.2.2

Add $0$ and $2$ .

$f (0) = 2$

Step 4.1.2.3

The final answer is $2$ .

$2$

Step 4.2

The point found by substituting $0$ in $f (x) = \frac{x^{5}}{10} + \frac{x^{4}}{8} + 2$ is $(0, 2)$ . This point can be an inflection point.

$(0, 2)$

Step 4.3

Substitute $- \frac{3}{4}$ in $f (x) = \frac{x^{5}}{10} + \frac{x^{4}}{8} + 2$ to find the value of $y$ .

Tap for more steps...

Step 4.3.1

Replace the variable $x$ with $- \frac{3}{4}$ in the expression.

$f (- \frac{3}{4}) = \frac{{(- \frac{3}{4})}^{5}}{10} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2

Simplify the result.

Tap for more steps...

Step 4.3.2.1

Simplify each term.

Tap for more steps...

Step 4.3.2.1.1

Simplify the numerator.

Tap for more steps...

Step 4.3.2.1.1.1

Apply the product rule to $- \frac{3}{4}$ .

$f (- \frac{3}{4}) = \frac{{(- 1)}^{5} {(\frac{3}{4})}^{5}}{10} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.1.2

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

$f (- \frac{3}{4}) = \frac{- {(\frac{3}{4})}^{5}}{10} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.1.3

Apply the product rule to $\frac{3}{4}$ .

$f (- \frac{3}{4}) = \frac{- \frac{3^{5}}{4^{5}}}{10} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.1.4

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

$f (- \frac{3}{4}) = \frac{- \frac{243}{4^{5}}}{10} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.1.5

Raise $4$ to the power of $5$ .

$f (- \frac{3}{4}) = \frac{- \frac{243}{1024}}{10} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{3}{4}) = - \frac{243}{1024} \cdot \frac{1}{10} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.3

Multiply $- \frac{243}{1024} \cdot \frac{1}{10}$ .

Tap for more steps...

Step 4.3.2.1.3.1

Multiply $\frac{1}{10}$ by $\frac{243}{1024}$ .

$f (- \frac{3}{4}) = - \frac{243}{10 \cdot 1024} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.3.2

Multiply $10$ by $1024$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{{(- \frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.4

Simplify the numerator.

Tap for more steps...

Step 4.3.2.1.4.1

Apply the product rule to $- \frac{3}{4}$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{{(- 1)}^{4} {(\frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.4.2

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

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{1 {(\frac{3}{4})}^{4}}{8} + 2$

Step 4.3.2.1.4.3

Apply the product rule to $\frac{3}{4}$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{1 (\frac{3^{4}}{4^{4}})}{8} + 2$

Step 4.3.2.1.4.4

Raise $3$ to the power of $4$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{1 (\frac{81}{4^{4}})}{8} + 2$

Step 4.3.2.1.4.5

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

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{1 (\frac{81}{256})}{8} + 2$

Step 4.3.2.1.4.6

Multiply $\frac{81}{256}$ by $1$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{\frac{81}{256}}{8} + 2$

Step 4.3.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81}{256} \cdot \frac{1}{8} + 2$

Step 4.3.2.1.6

Multiply $\frac{81}{256} \cdot \frac{1}{8}$ .

Tap for more steps...

Step 4.3.2.1.6.1

Multiply $\frac{81}{256}$ by $\frac{1}{8}$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81}{256 \cdot 8} + 2$

Step 4.3.2.1.6.2

Multiply $256$ by $8$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81}{2048} + 2$

Step 4.3.2.2

Find the common denominator.

Tap for more steps...

Step 4.3.2.2.1

Multiply $\frac{81}{2048}$ by $\frac{5}{5}$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81}{2048} \cdot \frac{5}{5} + 2$

Step 4.3.2.2.2

Multiply $\frac{81}{2048}$ by $\frac{5}{5}$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81 \cdot 5}{2048 \cdot 5} + 2$

Step 4.3.2.2.3

Write $2$ as a fraction with denominator $1$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81 \cdot 5}{2048 \cdot 5} + \frac{2}{1}$

Step 4.3.2.2.4

Multiply $\frac{2}{1}$ by $\frac{10240}{10240}$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81 \cdot 5}{2048 \cdot 5} + \frac{2}{1} \cdot \frac{10240}{10240}$

Step 4.3.2.2.5

Multiply $\frac{2}{1}$ by $\frac{10240}{10240}$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81 \cdot 5}{2048 \cdot 5} + \frac{2 \cdot 10240}{10240}$

Step 4.3.2.2.6

Reorder the factors of $2048 \cdot 5$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81 \cdot 5}{5 \cdot 2048} + \frac{2 \cdot 10240}{10240}$

Step 4.3.2.2.7

Multiply $5$ by $2048$ .

$f (- \frac{3}{4}) = - \frac{243}{10240} + \frac{81 \cdot 5}{10240} + \frac{2 \cdot 10240}{10240}$

Step 4.3.2.3

Combine the numerators over the common denominator.

$f (- \frac{3}{4}) = \frac{- 243 + 81 \cdot 5 + 2 \cdot 10240}{10240}$

Step 4.3.2.4

Simplify each term.

Tap for more steps...

Step 4.3.2.4.1

Multiply $81$ by $5$ .

$f (- \frac{3}{4}) = \frac{- 243 + 405 + 2 \cdot 10240}{10240}$

Step 4.3.2.4.2

Multiply $2$ by $10240$ .

$f (- \frac{3}{4}) = \frac{- 243 + 405 + 20480}{10240}$

Step 4.3.2.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.3.2.5.1

Add $- 243$ and $405$ .

$f (- \frac{3}{4}) = \frac{162 + 20480}{10240}$

Step 4.3.2.5.2

Add $162$ and $20480$ .

$f (- \frac{3}{4}) = \frac{20642}{10240}$

Step 4.3.2.5.3

Cancel the common factor of $20642$ and $10240$ .

Tap for more steps...

Step 4.3.2.5.3.1

Factor $2$ out of $20642$ .

$f (- \frac{3}{4}) = \frac{2 (10321)}{10240}$

Step 4.3.2.5.3.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.5.3.2.1

Factor $2$ out of $10240$ .

$f (- \frac{3}{4}) = \frac{2 \cdot 10321}{2 \cdot 5120}$

Step 4.3.2.5.3.2.2

Cancel the common factor.

$f (- \frac{3}{4}) = \frac{2 \cdot 10321}{2 \cdot 5120}$

Step 4.3.2.5.3.2.3

Rewrite the expression.

$f (- \frac{3}{4}) = \frac{10321}{5120}$

Step 4.3.2.6

The final answer is $\frac{10321}{5120}$ .

$\frac{10321}{5120}$

Step 4.4

The point found by substituting $- \frac{3}{4}$ in $f (x) = \frac{x^{5}}{10} + \frac{x^{4}}{8} + 2$ is $(- \frac{3}{4}, \frac{10321}{5120})$ . This point can be an inflection point.

$(- \frac{3}{4}, \frac{10321}{5120})$

Step 4.5

Determine the points that could be inflection points.

$(0, 2), (- \frac{3}{4}, \frac{10321}{5120})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - \frac{3}{4}) \cup (- \frac{3}{4}, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - \frac{3}{4})$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 0.85$ in the expression.

$f'' (- 0.85) = 2 {(- 0.85)}^{3} + \frac{3 {(- 0.85)}^{2}}{2}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Raise $- 0.85$ to the power of $3$ .

$f'' (- 0.85) = 2 \cdot - 0.614125 + \frac{3 {(- 0.85)}^{2}}{2}$

Step 6.2.1.2

Multiply $2$ by $- 0.614125$ .

$f'' (- 0.85) = - 1.22825 + \frac{3 {(- 0.85)}^{2}}{2}$

Step 6.2.1.3

Raise $- 0.85$ to the power of $2$ .

$f'' (- 0.85) = - 1.22825 + \frac{3 \cdot 0.7225}{2}$

Step 6.2.1.4

Multiply $3$ by $0.7225$ .

$f'' (- 0.85) = - 1.22825 + \frac{2.1675}{2}$

Step 6.2.1.5

Divide $2.1675$ by $2$ .

$f'' (- 0.85) = - 1.22825 + 1.08375$

Step 6.2.2

Add $- 1.22825$ and $1.08375$ .

$f'' (- 0.85) = - 0.1445$

Step 6.2.3

The final answer is $- 0.1445$ .

$- 0.1445$

Step 6.3

At $- 0.85$ , the second derivative is $- 0.1445$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \frac{3}{4})$

Decreasing on $(- \infty, - \frac{3}{4})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- \frac{3}{4}, 0)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $- 0.375$ in the expression.

$f'' (- 0.375) = 2 {(- 0.375)}^{3} + \frac{3 {(- 0.375)}^{2}}{2}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

Raise $- 0.375$ to the power of $3$ .

$f'' (- 0.375) = 2 \cdot - 0.05273437 + \frac{3 {(- 0.375)}^{2}}{2}$

Step 7.2.1.2

Multiply $2$ by $- 0.05273437$ .

$f'' (- 0.375) = - 0.10546875 + \frac{3 {(- 0.375)}^{2}}{2}$

Step 7.2.1.3

Raise $- 0.375$ to the power of $2$ .

$f'' (- 0.375) = - 0.10546875 + \frac{3 \cdot 0.140625}{2}$

Step 7.2.1.4

Multiply $3$ by $0.140625$ .

$f'' (- 0.375) = - 0.10546875 + \frac{0.421875}{2}$

Step 7.2.1.5

Divide $0.421875$ by $2$ .

$f'' (- 0.375) = - 0.10546875 + 0.2109375$

Step 7.2.2

Add $- 0.10546875$ and $0.2109375$ .

$f'' (- 0.375) = 0.10546875$

Step 7.2.3

The final answer is $0.10546875$ .

$0.10546875$

Step 7.3

At $- 0.375$ , the second derivative is $0.10546875$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{3}{4}, 0)$ .

Increasing on $(- \frac{3}{4}, 0)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = 2 {(0.1)}^{3} + \frac{3 {(0.1)}^{2}}{2}$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify each term.

Tap for more steps...

Step 8.2.1.1

Raise $0.1$ to the power of $3$ .

$f'' (0.1) = 2 \cdot 0.001 + \frac{3 {(0.1)}^{2}}{2}$

Step 8.2.1.2

Multiply $2$ by $0.001$ .

$f'' (0.1) = 0.002 + \frac{3 {(0.1)}^{2}}{2}$

Step 8.2.1.3

Raise $0.1$ to the power of $2$ .

$f'' (0.1) = 0.002 + \frac{3 \cdot 0.01}{2}$

Step 8.2.1.4

Multiply $3$ by $0.01$ .

$f'' (0.1) = 0.002 + \frac{0.03}{2}$

Step 8.2.1.5

Divide $0.03$ by $2$ .

$f'' (0.1) = 0.002 + 0.015$

Step 8.2.2

Add $0.002$ and $0.015$ .

$f'' (0.1) = 0.017$

Step 8.2.3

The final answer is $0.017$ .

$0.017$

Step 8.3

At $0.1$ , the second derivative is $0.017$ . Since this is positive, the second derivative is increasing on the interval $(0, \infty)$ .

Increasing on $(0, \infty)$ since $f'' (x) > 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(- \frac{3}{4}, \frac{10321}{5120})$ .

$(- \frac{3}{4}, \frac{10321}{5120})$

Step 10