Calculus Examples

$f (x) = 2 x^{\frac{5}{2}} - x^{2}$

Step 1

Find the first derivative of the function.

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Step 1.1

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

$\frac{d}{d x} [2 x^{\frac{5}{2}}] + \frac{d}{d x} [- x^{2}]$

Step 1.2

Evaluate $\frac{d}{d x} [2 x^{\frac{5}{2}}]$ .

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Step 1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{5}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$2 \frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- x^{2}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

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

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 1.2.4

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

$2 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 1.2.5

Combine the numerators over the common denominator.

$2 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

$2 (\frac{5}{2} x^{\frac{5 - 2}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 1.2.6.2

Subtract $2$ from $5$ .

$2 (\frac{5}{2} x^{\frac{3}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 1.2.7

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$2 \frac{5 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- x^{2}]$

Step 1.2.8

Combine $2$ and $\frac{5 x^{\frac{3}{2}}}{2}$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- x^{2}]$

Step 1.2.9

Multiply $5$ by $2$ .

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

Step 1.2.10

Factor $2$ out of $10 x^{\frac{3}{2}}$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- x^{2}]$

Step 1.2.11

Cancel the common factors.

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Step 1.2.11.1

Factor $2$ out of $2$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2 (1)} + \frac{d}{d x} [- x^{2}]$

Step 1.2.11.2

Cancel the common factor.

$\frac{2 (5 x^{\frac{3}{2}})}{2 \cdot 1} + \frac{d}{d x} [- x^{2}]$

Step 1.2.11.3

Rewrite the expression.

$\frac{5 x^{\frac{3}{2}}}{1} + \frac{d}{d x} [- x^{2}]$

Step 1.2.11.4

Divide $5 x^{\frac{3}{2}}$ by $1$ .

$5 x^{\frac{3}{2}} + \frac{d}{d x} [- x^{2}]$

Step 1.3

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

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Step 1.3.1

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

$5 x^{\frac{3}{2}} - \frac{d}{d x} [x^{2}]$

Step 1.3.2

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

$5 x^{\frac{3}{2}} - (2 x)$

Step 1.3.3

Multiply $2$ by $- 1$ .

$5 x^{\frac{3}{2}} - 2 x$

Step 1.4

Reorder terms.

$- 2 x + 5 x^{\frac{3}{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 2 x + 5 x^{\frac{3}{2}}$ with respect to $x$ is $\frac{d}{d x} [- 2 x] + \frac{d}{d x} [5 x^{\frac{3}{2}}]$ .

$f'' (x) = \frac{d}{d x} (- 2 x) + \frac{d}{d x} (5 x^{\frac{3}{2}})$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = - 2 \frac{d}{d x} x + \frac{d}{d x} (5 x^{\frac{3}{2}})$

Step 2.2.2

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

$f'' (x) = - 2 \cdot 1 + \frac{d}{d x} (5 x^{\frac{3}{2}})$

Step 2.2.3

Multiply $- 2$ by $1$ .

$f'' (x) = - 2 + \frac{d}{d x} (5 x^{\frac{3}{2}})$

Step 2.3

Evaluate $\frac{d}{d x} [5 x^{\frac{3}{2}}]$ .

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Step 2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{3}{2}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$f'' (x) = - 2 + 5 \frac{d}{d x} (x^{\frac{3}{2}})$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$f'' (x) = - 2 + 5 (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 2.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - 2 + 5 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 2.3.4

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

$f'' (x) = - 2 + 5 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.3.5

Combine the numerators over the common denominator.

$f'' (x) = - 2 + 5 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $2$ .

$f'' (x) = - 2 + 5 (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 2.3.6.2

Subtract $2$ from $3$ .

$f'' (x) = - 2 + 5 (\frac{3}{2} x^{\frac{1}{2}})$

Step 2.3.7

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

$f'' (x) = - 2 + 5 (\frac{3 x^{\frac{1}{2}}}{2})$

Step 2.3.8

Combine $5$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$f'' (x) = - 2 + \frac{5 (3 x^{\frac{1}{2}})}{2}$

Step 2.3.9

Multiply $3$ by $5$ .

$f'' (x) = - 2 + \frac{15 x^{\frac{1}{2}}}{2}$

Step 3

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

$- 2 x + 5 x^{\frac{3}{2}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

$\frac{d}{d x} [2 x^{\frac{5}{2}}] + \frac{d}{d x} [- x^{2}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [2 x^{\frac{5}{2}}]$ .

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Step 4.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{5}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$2 \frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

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

Step 4.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.4

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

$2 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.5

Combine the numerators over the common denominator.

$2 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $2$ .

$2 (\frac{5}{2} x^{\frac{5 - 2}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.6.2

Subtract $2$ from $5$ .

$2 (\frac{5}{2} x^{\frac{3}{2}}) + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.7

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$2 \frac{5 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.8

Combine $2$ and $\frac{5 x^{\frac{3}{2}}}{2}$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.9

Multiply $5$ by $2$ .

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

Step 4.1.2.10

Factor $2$ out of $10 x^{\frac{3}{2}}$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.11

Cancel the common factors.

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Step 4.1.2.11.1

Factor $2$ out of $2$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2 (1)} + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.11.2

Cancel the common factor.

$\frac{2 (5 x^{\frac{3}{2}})}{2 \cdot 1} + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.11.3

Rewrite the expression.

$\frac{5 x^{\frac{3}{2}}}{1} + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.11.4

Divide $5 x^{\frac{3}{2}}$ by $1$ .

$5 x^{\frac{3}{2}} + \frac{d}{d x} [- x^{2}]$

Step 4.1.3

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

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Step 4.1.3.1

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

$5 x^{\frac{3}{2}} - \frac{d}{d x} [x^{2}]$

Step 4.1.3.2

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

$5 x^{\frac{3}{2}} - (2 x)$

Step 4.1.3.3

Multiply $2$ by $- 1$ .

$5 x^{\frac{3}{2}} - 2 x$

Step 4.1.4

Reorder terms.

$f' (x) = - 2 x + 5 x^{\frac{3}{2}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 2 x + 5 x^{\frac{3}{2}}$ .

$- 2 x + 5 x^{\frac{3}{2}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 2 x + 5 x^{\frac{3}{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 2 x + 5 x^{\frac{3}{2}} = 0$

Step 5.2

Factor $- x$ out of $- 2 x + 5 x^{\frac{3}{2}}$ .

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Step 5.2.1

Factor $- x$ out of $- 2 x$ .

$- x \cdot 2 + 5 x^{\frac{3}{2}} = 0$

Step 5.2.2

Factor $- x$ out of $5 x^{\frac{3}{2}}$ .

$- x \cdot 2 - x (- 5 x^{\frac{1}{2}}) = 0$

Step 5.2.3

Factor $- x$ out of $- x (2) - x (- 5 x^{\frac{1}{2}})$ .

$- x (2 - 5 x^{\frac{1}{2}}) = 0$

Step 5.3

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

$x = 0$

$2 - 5 x^{\frac{1}{2}} = 0$

Step 5.4

Set $x$ equal to $0$ .

$x = 0$

Step 5.5

Set $2 - 5 x^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $2 - 5 x^{\frac{1}{2}}$ equal to $0$ .

$2 - 5 x^{\frac{1}{2}} = 0$

Step 5.5.2

Solve $2 - 5 x^{\frac{1}{2}} = 0$ for $x$ .

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Step 5.5.2.1

Subtract $2$ from both sides of the equation.

$- 5 x^{\frac{1}{2}} = - 2$

Step 5.5.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(- 5 x^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.5.2.3

Simplify the exponent.

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Step 5.5.2.3.1

Simplify the left side.

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Step 5.5.2.3.1.1

Simplify ${(- 5 x^{\frac{1}{2}})}^{2}$ .

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Step 5.5.2.3.1.1.1

Apply the product rule to $- 5 x^{\frac{1}{2}}$ .

${(- 5)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.5.2.3.1.1.2

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

$25 {(x^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.5.2.3.1.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 5.5.2.3.1.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$25 x^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Step 5.5.2.3.1.1.3.2

Cancel the common factor of $2$ .

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Step 5.5.2.3.1.1.3.2.1

Cancel the common factor.

$25 x^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Step 5.5.2.3.1.1.3.2.2

Rewrite the expression.

$25 x^{1} = {(- 2)}^{2}$

Step 5.5.2.3.1.1.4

Simplify.

$25 x = {(- 2)}^{2}$

Step 5.5.2.3.2

Simplify the right side.

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Step 5.5.2.3.2.1

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

$25 x = 4$

Step 5.5.2.4

Divide each term in $25 x = 4$ by $25$ and simplify.

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Step 5.5.2.4.1

Divide each term in $25 x = 4$ by $25$ .

$\frac{25 x}{25} = \frac{4}{25}$

Step 5.5.2.4.2

Simplify the left side.

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Step 5.5.2.4.2.1

Cancel the common factor of $25$ .

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Step 5.5.2.4.2.1.1

Cancel the common factor.

$\frac{25 x}{25} = \frac{4}{25}$

Step 5.5.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{25}$

Step 5.6

The final solution is all the values that make $- x (2 - 5 x^{\frac{1}{2}}) = 0$ true.

$x = 0, \frac{4}{25}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- 2 x + 5 \sqrt{x^{3}}$

Step 6.2

Set the radicand in $\sqrt{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 6.3.2

Simplify the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 6.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 6.3.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 6.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

Step 7

Critical points to evaluate.

$x = 0, \frac{4}{25}$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2 + \frac{15 {(0)}^{\frac{1}{2}}}{2}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Simplify the numerator.

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Step 9.1.1.1

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

$- 2 + \frac{15 \cdot {(0^{2})}^{\frac{1}{2}}}{2}$

Step 9.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 2 + \frac{15 \cdot 0^{2 (\frac{1}{2})}}{2}$

Step 9.1.1.3

Cancel the common factor of $2$ .

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Step 9.1.1.3.1

Cancel the common factor.

$- 2 + \frac{15 \cdot 0^{2 (\frac{1}{2})}}{2}$

Step 9.1.1.3.2

Rewrite the expression.

$- 2 + \frac{15 \cdot 0^{1}}{2}$

Step 9.1.1.4

Evaluate the exponent.

$- 2 + \frac{15 \cdot 0}{2}$

Step 9.1.2

Multiply $15$ by $0$ .

$- 2 + \frac{0}{2}$

Step 9.1.3

Divide $0$ by $2$ .

$- 2 + 0$

Step 9.2

Add $- 2$ and $0$ .

$- 2$

Step 10

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 11

Find the y-value when $x = 0$ .

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Step 11.1

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

$f (0) = 2 {(0)}^{\frac{5}{2}} - {(0)}^{2}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

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

$f (0) = 2 {(0^{2})}^{\frac{5}{2}} - {(0)}^{2}$

Step 11.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (0) = 2 \cdot 0^{2 (\frac{5}{2})} - {(0)}^{2}$

Step 11.2.1.3

Cancel the common factor of $2$ .

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Step 11.2.1.3.1

Cancel the common factor.

$f (0) = 2 \cdot 0^{2 (\frac{5}{2})} - {(0)}^{2}$

Step 11.2.1.3.2

Rewrite the expression.

$f (0) = 2 \cdot 0^{5} - {(0)}^{2}$

Step 11.2.1.4

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

$f (0) = 2 \cdot 0 - {(0)}^{2}$

Step 11.2.1.5

Multiply $2$ by $0$ .

$f (0) = 0 - {(0)}^{2}$

Step 11.2.1.6

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

$f (0) = 0 - 0$

Step 11.2.1.7

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0$

Step 11.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 11.2.3

The final answer is $0$ .

$y = 0$

Step 12

Evaluate the second derivative at $x = \frac{4}{25}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2 + \frac{15 {(\frac{4}{25})}^{\frac{1}{2}}}{2}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Simplify the numerator.

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Step 13.1.1.1

Apply the product rule to $\frac{4}{25}$ .

$- 2 + \frac{15 \frac{4^{\frac{1}{2}}}{25^{\frac{1}{2}}}}{2}$

Step 13.1.1.2

Simplify the numerator.

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Step 13.1.1.2.1

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

$- 2 + \frac{15 \frac{{(2^{2})}^{\frac{1}{2}}}{25^{\frac{1}{2}}}}{2}$

Step 13.1.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 2 + \frac{15 \frac{2^{2 (\frac{1}{2})}}{25^{\frac{1}{2}}}}{2}$

Step 13.1.1.2.3

Cancel the common factor of $2$ .

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Step 13.1.1.2.3.1

Cancel the common factor.

$- 2 + \frac{15 \frac{2^{2 (\frac{1}{2})}}{25^{\frac{1}{2}}}}{2}$

Step 13.1.1.2.3.2

Rewrite the expression.

$- 2 + \frac{15 \frac{2^{1}}{25^{\frac{1}{2}}}}{2}$

Step 13.1.1.2.4

Evaluate the exponent.

$- 2 + \frac{15 \frac{2}{25^{\frac{1}{2}}}}{2}$

Step 13.1.1.3

Simplify the denominator.

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Step 13.1.1.3.1

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

$- 2 + \frac{15 \frac{2}{{(5^{2})}^{\frac{1}{2}}}}{2}$

Step 13.1.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 2 + \frac{15 \frac{2}{5^{2 (\frac{1}{2})}}}{2}$

Step 13.1.1.3.3

Cancel the common factor of $2$ .

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Step 13.1.1.3.3.1

Cancel the common factor.

$- 2 + \frac{15 \frac{2}{5^{2 (\frac{1}{2})}}}{2}$

Step 13.1.1.3.3.2

Rewrite the expression.

$- 2 + \frac{15 \frac{2}{5^{1}}}{2}$

Step 13.1.1.3.4

Evaluate the exponent.

$- 2 + \frac{15 (\frac{2}{5})}{2}$

Step 13.1.2

Combine $15$ and $\frac{2}{5}$ .

$- 2 + \frac{\frac{15 \cdot 2}{5}}{2}$

Step 13.1.3

Multiply $15$ by $2$ .

$- 2 + \frac{\frac{30}{5}}{2}$

Step 13.1.4

Divide $30$ by $5$ .

$- 2 + \frac{6}{2}$

Step 13.1.5

Divide $6$ by $2$ .

$- 2 + 3$

Step 13.2

Add $- 2$ and $3$ .

$1$

Step 14

$x = \frac{4}{25}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{4}{25}$ is a local minimum

Step 15

Find the y-value when $x = \frac{4}{25}$ .

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Step 15.1

Replace the variable $x$ with $\frac{4}{25}$ in the expression.

$f (\frac{4}{25}) = 2 {(\frac{4}{25})}^{\frac{5}{2}} - {(\frac{4}{25})}^{2}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Apply the product rule to $\frac{4}{25}$ .

$f (\frac{4}{25}) = 2 (\frac{4^{\frac{5}{2}}}{25^{\frac{5}{2}}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.2

Simplify the numerator.

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Step 15.2.1.2.1

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

$f (\frac{4}{25}) = 2 (\frac{{(2^{2})}^{\frac{5}{2}}}{25^{\frac{5}{2}}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{4}{25}) = 2 (\frac{2^{2 (\frac{5}{2})}}{25^{\frac{5}{2}}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.2.3

Cancel the common factor of $2$ .

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Step 15.2.1.2.3.1

Cancel the common factor.

$f (\frac{4}{25}) = 2 (\frac{2^{2 (\frac{5}{2})}}{25^{\frac{5}{2}}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.2.3.2

Rewrite the expression.

$f (\frac{4}{25}) = 2 (\frac{2^{5}}{25^{\frac{5}{2}}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.2.4

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

$f (\frac{4}{25}) = 2 (\frac{32}{25^{\frac{5}{2}}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.3

Simplify the denominator.

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Step 15.2.1.3.1

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

$f (\frac{4}{25}) = 2 (\frac{32}{{(5^{2})}^{\frac{5}{2}}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{4}{25}) = 2 (\frac{32}{5^{2 (\frac{5}{2})}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.3.3

Cancel the common factor of $2$ .

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Step 15.2.1.3.3.1

Cancel the common factor.

$f (\frac{4}{25}) = 2 (\frac{32}{5^{2 (\frac{5}{2})}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.3.3.2

Rewrite the expression.

$f (\frac{4}{25}) = 2 (\frac{32}{5^{5}}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.3.4

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

$f (\frac{4}{25}) = 2 (\frac{32}{3125}) - {(\frac{4}{25})}^{2}$

Step 15.2.1.4

Multiply $2 (\frac{32}{3125})$ .

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Step 15.2.1.4.1

Combine $2$ and $\frac{32}{3125}$ .

$f (\frac{4}{25}) = \frac{2 \cdot 32}{3125} - {(\frac{4}{25})}^{2}$

Step 15.2.1.4.2

Multiply $2$ by $32$ .

$f (\frac{4}{25}) = \frac{64}{3125} - {(\frac{4}{25})}^{2}$

Step 15.2.1.5

Apply the product rule to $\frac{4}{25}$ .

$f (\frac{4}{25}) = \frac{64}{3125} - \frac{4^{2}}{25^{2}}$

Step 15.2.1.6

Raise $4$ to the power of $2$ .

$f (\frac{4}{25}) = \frac{64}{3125} - \frac{16}{25^{2}}$

Step 15.2.1.7

Raise $25$ to the power of $2$ .

$f (\frac{4}{25}) = \frac{64}{3125} - \frac{16}{625}$

Step 15.2.2

To write $- \frac{16}{625}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{4}{25}) = \frac{64}{3125} - \frac{16}{625} \cdot \frac{5}{5}$

Step 15.2.3

Write each expression with a common denominator of $3125$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.2.3.1

Multiply $\frac{16}{625}$ by $\frac{5}{5}$ .

$f (\frac{4}{25}) = \frac{64}{3125} - \frac{16 \cdot 5}{625 \cdot 5}$

Step 15.2.3.2

Multiply $625$ by $5$ .

$f (\frac{4}{25}) = \frac{64}{3125} - \frac{16 \cdot 5}{3125}$

Step 15.2.4

Combine the numerators over the common denominator.

$f (\frac{4}{25}) = \frac{64 - 16 \cdot 5}{3125}$

Step 15.2.5

Simplify the numerator.

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Step 15.2.5.1

Multiply $- 16$ by $5$ .

$f (\frac{4}{25}) = \frac{64 - 80}{3125}$

Step 15.2.5.2

Subtract $80$ from $64$ .

$f (\frac{4}{25}) = \frac{- 16}{3125}$

Step 15.2.6

Move the negative in front of the fraction.

$f (\frac{4}{25}) = - \frac{16}{3125}$

Step 15.2.7

The final answer is $- \frac{16}{3125}$ .

$y = - \frac{16}{3125}$

Step 16

These are the local extrema for $f (x) = 2 x^{\frac{5}{2}} - x^{2}$ .

$(0, 0)$ is a local maxima

$(\frac{4}{25}, - \frac{16}{3125})$ is a local minima

Step 17