Calculus Examples

$f (x) = \sin (x^{2})$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x^{2}$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x^{2}]$

Step 1.1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\cos (u) \frac{d}{d x} [x^{2}]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 1.2

Differentiate using the Power Rule.

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Step 1.2.1

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

$\cos (x^{2}) (2 x)$

Step 1.2.2

Reorder the factors of $\cos (x^{2}) (2 x)$ .

$2 x \cos (x^{2})$

Step 2

Find the second derivative of the function.

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Step 2.1

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

$f'' (x) = 2 \frac{d}{d x} (x \cos (x^{2}))$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos (x^{2})$ .

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = x^{2}$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$f'' (x) = 2 (x (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (x^{2})) + \cos (x^{2}) \frac{d}{d x} (x))$

Step 2.3.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$f'' (x) = 2 (x (- \sin (u) \frac{d}{d x} x^{2}) + \cos (x^{2}) \frac{d}{d x} (x))$

Step 2.3.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2.4

Differentiate using the Power Rule.

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Step 2.4.1

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

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

Step 2.4.2

Multiply $2$ by $- 1$ .

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

Step 2.5

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

$f'' (x) = 2 (x \cdot x (- 2 \sin (x^{2})) + \cos (x^{2}) \frac{d}{d x} (x))$

Step 2.6

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

$f'' (x) = 2 (x \cdot x (- 2 \sin (x^{2})) + \cos (x^{2}) \frac{d}{d x} (x))$

Step 2.7

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

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

Step 2.8

Add $1$ and $1$ .

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

Step 2.9

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

$f'' (x) = 2 (x^{2} (- 2 \sin (x^{2})) + \cos (x^{2}) \cdot 1)$

Step 2.10

Multiply $\cos (x^{2})$ by $1$ .

$f'' (x) = 2 (x^{2} (- 2 \sin (x^{2})) + \cos (x^{2}))$

Step 2.11

Simplify.

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Step 2.11.1

Apply the distributive property.

$f'' (x) = 2 (x^{2} (- 2 \sin (x^{2}))) + 2 \cos (x^{2})$

Step 2.11.2

Multiply $- 2$ by $2$ .

$f'' (x) = - 4 x^{2} \sin (x^{2}) + 2 \cos (x^{2})$

Step 3

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

$2 x \cos (x^{2}) = 0$

Step 4

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

$x = 0$

$\cos (x^{2}) = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $\cos (x^{2})$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\cos (x^{2})$ equal to $0$ .

$\cos (x^{2}) = 0$

Step 6.2

Solve $\cos (x^{2}) = 0$ for $x$ .

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Step 6.2.1

Substitute $u$ for $x^{2}$ .

$\cos (u) = 0$

Step 6.2.2

Take the inverse cosine of both sides of the equation to extract $u$ from inside the cosine.

$u = \arccos (0)$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$u = \frac{π}{2}$

Step 6.2.4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$u = 2 π - \frac{π}{2}$

Step 6.2.5

Simplify $2 π - \frac{π}{2}$ .

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Step 6.2.5.1

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

$u = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.2.5.2

Combine fractions.

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Step 6.2.5.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$u = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 6.2.5.2.2

Combine the numerators over the common denominator.

$u = \frac{2 π \cdot 2 - π}{2}$

Step 6.2.5.3

Simplify the numerator.

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Step 6.2.5.3.1

Multiply $2$ by $2$ .

$u = \frac{4 π - π}{2}$

Step 6.2.5.3.2

Subtract $π$ from $4 π$ .

$u = \frac{3 π}{2}$

Step 6.2.6

The solution to the equation $u = \frac{π}{2}$ .

$u = \frac{π}{2}, \frac{3 π}{2}$

Step 6.2.7

Substitute $x^{2}$ for $u$ and solve $x^{2} = \frac{π}{2}$

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Step 6.2.7.1

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

$x = \pm \sqrt{\frac{π}{2}}$

Step 6.2.7.2

Simplify $\pm \sqrt{\frac{π}{2}}$ .

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Step 6.2.7.2.1

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

$x = \pm \frac{\sqrt{π}}{\sqrt{2}}$

Step 6.2.7.2.2

Multiply $\frac{\sqrt{π}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{π}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.2.7.2.3

Combine and simplify the denominator.

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Step 6.2.7.2.3.1

Multiply $\frac{\sqrt{π}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 6.2.7.2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.2.7.2.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.2.7.2.3.4

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

$x = \pm \frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.2.7.2.3.5

Add $1$ and $1$ .

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

Step 6.2.7.2.3.6

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

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Step 6.2.7.2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{π} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.2.7.2.3.6.2

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

$x = \pm \frac{\sqrt{π} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.2.7.2.3.6.3

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

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

Step 6.2.7.2.3.6.4

Cancel the common factor of $2$ .

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Step 6.2.7.2.3.6.4.1

Cancel the common factor.

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

Step 6.2.7.2.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{π} \sqrt{2}}{2^{1}}$

Step 6.2.7.2.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{π} \sqrt{2}}{2}$

Step 6.2.7.2.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{π \cdot 2}}{2}$

Step 6.2.7.2.5

Reorder factors in $\pm \frac{\sqrt{π \cdot 2}}{2}$ .

$x = \pm \frac{\sqrt{2 π}}{2}$

Step 6.2.7.3

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

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Step 6.2.7.3.1

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

$x = \frac{\sqrt{2 π}}{2}$

Step 6.2.7.3.2

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

$x = - \frac{\sqrt{2 π}}{2}$

Step 6.2.7.3.3

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

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

Step 6.2.8

Substitute $x^{2}$ for $u$ and solve $x^{2} = \frac{3 π}{2}$

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Step 6.2.8.1

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

$x = \pm \sqrt{\frac{3 π}{2}}$

Step 6.2.8.2

Simplify $\pm \sqrt{\frac{3 π}{2}}$ .

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Step 6.2.8.2.1

Rewrite $\sqrt{\frac{3 π}{2}}$ as $\frac{\sqrt{3 π}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{3 π}}{\sqrt{2}}$

Step 6.2.8.2.2

Multiply $\frac{\sqrt{3 π}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 6.2.8.2.3

Combine and simplify the denominator.

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Step 6.2.8.2.3.1

Multiply $\frac{\sqrt{3 π}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 6.2.8.2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3 π} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.2.8.2.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3 π} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.2.8.2.3.4

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

$x = \pm \frac{\sqrt{3 π} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.2.8.2.3.5

Add $1$ and $1$ .

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

Step 6.2.8.2.3.6

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

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Step 6.2.8.2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3 π} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.2.8.2.3.6.2

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

$x = \pm \frac{\sqrt{3 π} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.2.8.2.3.6.3

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

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

Step 6.2.8.2.3.6.4

Cancel the common factor of $2$ .

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Step 6.2.8.2.3.6.4.1

Cancel the common factor.

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

Step 6.2.8.2.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3 π} \sqrt{2}}{2^{1}}$

Step 6.2.8.2.3.6.5

Evaluate the exponent.

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

Step 6.2.8.2.4

Simplify the numerator.

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Step 6.2.8.2.4.1

Combine using the product rule for radicals.

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

Step 6.2.8.2.4.2

Multiply $2$ by $3$ .

$x = \pm \frac{\sqrt{6 π}}{2}$

Step 6.2.8.3

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

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Step 6.2.8.3.1

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

$x = \frac{\sqrt{6 π}}{2}$

Step 6.2.8.3.2

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

$x = - \frac{\sqrt{6 π}}{2}$

Step 6.2.8.3.3

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

$x = \frac{\sqrt{6 π}}{2}, - \frac{\sqrt{6 π}}{2}$

Step 7

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

$x = 0, \frac{\sqrt{6 π}}{2}, - \frac{\sqrt{6 π}}{2}$

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.

$- 4 {(0)}^{2} \sin ({(0)}^{2}) + 2 \cos ({(0)}^{2})$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$- 4 \cdot 0 \sin ({(0)}^{2}) + 2 \cos ({(0)}^{2})$

Step 9.1.2

Multiply $- 4$ by $0$ .

$0 \sin ({(0)}^{2}) + 2 \cos ({(0)}^{2})$

Step 9.1.3

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

$0 \sin (0) + 2 \cos ({(0)}^{2})$

Step 9.1.4

The exact value of $\sin (0)$ is $0$ .

$0 \cdot 0 + 2 \cos ({(0)}^{2})$

Step 9.1.5

Multiply $0$ by $0$ .

$0 + 2 \cos ({(0)}^{2})$

Step 9.1.6

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

$0 + 2 \cos (0)$

Step 9.1.7

The exact value of $\cos (0)$ is $1$ .

$0 + 2 \cdot 1$

Step 9.1.8

Multiply $2$ by $1$ .

$0 + 2$

Step 9.2

Add $0$ and $2$ .

$2$

Step 10

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

$x = 0$ is a local minimum

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.

$f (0) = \sin ({(0)}^{2})$

Step 11.2

Simplify the result.

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Step 11.2.1

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

$f (0) = \sin (0)$

Step 11.2.2

The exact value of $\sin (0)$ is $0$ .

$f (0) = 0$

Step 11.2.3

The final answer is $0$ .

$y = 0$

Step 12

Evaluate the second derivative at $x = \frac{\sqrt{6 π}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 4 {(\frac{\sqrt{6 π}}{2})}^{2} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the product rule to $\frac{\sqrt{6 π}}{2}$ .

$- 4 \frac{{\sqrt{6 π}}^{2}}{2^{2}} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.2

Rewrite ${\sqrt{6 π}}^{2}$ as $6 π$ .

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Step 13.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 π}$ as ${(6 π)}^{\frac{1}{2}}$ .

$- 4 \frac{{({(6 π)}^{\frac{1}{2}})}^{2}}{2^{2}} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.2.2

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

$- 4 \frac{{(6 π)}^{\frac{1}{2} \cdot 2}}{2^{2}} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.2.3

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

$- 4 \frac{{(6 π)}^{\frac{2}{2}}}{2^{2}} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.2.4

Cancel the common factor of $2$ .

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Step 13.1.2.4.1

Cancel the common factor.

$- 4 \frac{{(6 π)}^{\frac{2}{2}}}{2^{2}} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.2.4.2

Rewrite the expression.

$- 4 \frac{{(6 π)}^{1}}{2^{2}} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.2.5

Simplify.

$- 4 \frac{6 π}{2^{2}} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.3

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

$- 4 \frac{6 π}{4} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.4

Cancel the common factor of $4$ .

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Step 13.1.4.1

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{6 π}{4} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.4.2

Cancel the common factor.

$4 \cdot - 1 \frac{6 π}{4} \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.4.3

Rewrite the expression.

$- 1 (6 π) \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.5

Multiply $6$ by $- 1$ .

$- 6 π \sin ({(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.6

Apply the product rule to $\frac{\sqrt{6 π}}{2}$ .

$- 6 π \sin (\frac{{\sqrt{6 π}}^{2}}{2^{2}}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.7

Rewrite ${\sqrt{6 π}}^{2}$ as $6 π$ .

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Step 13.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 π}$ as ${(6 π)}^{\frac{1}{2}}$ .

$- 6 π \sin (\frac{{({(6 π)}^{\frac{1}{2}})}^{2}}{2^{2}}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.7.2

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

$- 6 π \sin (\frac{{(6 π)}^{\frac{1}{2} \cdot 2}}{2^{2}}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.7.3

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

$- 6 π \sin (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.7.4

Cancel the common factor of $2$ .

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Step 13.1.7.4.1

Cancel the common factor.

$- 6 π \sin (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.7.4.2

Rewrite the expression.

$- 6 π \sin (\frac{{(6 π)}^{1}}{2^{2}}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.7.5

Simplify.

$- 6 π \sin (\frac{6 π}{2^{2}}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.8

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

$- 6 π \sin (\frac{6 π}{4}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.9

Cancel the common factor of $6$ and $4$ .

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Step 13.1.9.1

Factor $2$ out of $6 π$ .

$- 6 π \sin (\frac{2 (3 π)}{4}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.9.2

Cancel the common factors.

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Step 13.1.9.2.1

Factor $2$ out of $4$ .

$- 6 π \sin (\frac{2 (3 π)}{2 (2)}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.9.2.2

Cancel the common factor.

$- 6 π \sin (\frac{2 (3 π)}{2 \cdot 2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.9.2.3

Rewrite the expression.

$- 6 π \sin (\frac{3 π}{2}) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.10

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- 6 π (- \sin (\frac{π}{2})) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.11

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- 6 π (- 1 \cdot 1) + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.12

Multiply $- 1$ by $1$ .

$- 6 π \cdot - 1 + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.13

Multiply $- 1$ by $- 6$ .

$6 π + 2 \cos ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 13.1.14

Apply the product rule to $\frac{\sqrt{6 π}}{2}$ .

$6 π + 2 \cos (\frac{{\sqrt{6 π}}^{2}}{2^{2}})$

Step 13.1.15

Rewrite ${\sqrt{6 π}}^{2}$ as $6 π$ .

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Step 13.1.15.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 π}$ as ${(6 π)}^{\frac{1}{2}}$ .

$6 π + 2 \cos (\frac{{({(6 π)}^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 13.1.15.2

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

$6 π + 2 \cos (\frac{{(6 π)}^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 13.1.15.3

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

$6 π + 2 \cos (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}})$

Step 13.1.15.4

Cancel the common factor of $2$ .

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Step 13.1.15.4.1

Cancel the common factor.

$6 π + 2 \cos (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}})$

Step 13.1.15.4.2

Rewrite the expression.

$6 π + 2 \cos (\frac{{(6 π)}^{1}}{2^{2}})$

Step 13.1.15.5

Simplify.

$6 π + 2 \cos (\frac{6 π}{2^{2}})$

Step 13.1.16

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

$6 π + 2 \cos (\frac{6 π}{4})$

Step 13.1.17

Cancel the common factor of $6$ and $4$ .

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Step 13.1.17.1

Factor $2$ out of $6 π$ .

$6 π + 2 \cos (\frac{2 (3 π)}{4})$

Step 13.1.17.2

Cancel the common factors.

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Step 13.1.17.2.1

Factor $2$ out of $4$ .

$6 π + 2 \cos (\frac{2 (3 π)}{2 (2)})$

Step 13.1.17.2.2

Cancel the common factor.

$6 π + 2 \cos (\frac{2 (3 π)}{2 \cdot 2})$

Step 13.1.17.2.3

Rewrite the expression.

$6 π + 2 \cos (\frac{3 π}{2})$

Step 13.1.18

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$6 π + 2 \cos (\frac{π}{2})$

Step 13.1.19

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$6 π + 2 \cdot 0$

Step 13.1.20

Multiply $2$ by $0$ .

$6 π + 0$

Step 13.2

Add $6 π$ and $0$ .

$6 π$

Step 14

$x = \frac{\sqrt{6 π}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{\sqrt{6 π}}{2}$ is a local minimum

Step 15

Find the y-value when $x = \frac{\sqrt{6 π}}{2}$ .

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Step 15.1

Replace the variable $x$ with $\frac{\sqrt{6 π}}{2}$ in the expression.

$f (\frac{\sqrt{6 π}}{2}) = \sin ({(\frac{\sqrt{6 π}}{2})}^{2})$

Step 15.2

Simplify the result.

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Step 15.2.1

Apply the product rule to $\frac{\sqrt{6 π}}{2}$ .

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{{\sqrt{6 π}}^{2}}{2^{2}})$

Step 15.2.2

Rewrite ${\sqrt{6 π}}^{2}$ as $6 π$ .

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Step 15.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 π}$ as ${(6 π)}^{\frac{1}{2}}$ .

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{{({(6 π)}^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 15.2.2.2

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

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{{(6 π)}^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 15.2.2.3

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

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}})$

Step 15.2.2.4

Cancel the common factor of $2$ .

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Step 15.2.2.4.1

Cancel the common factor.

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}})$

Step 15.2.2.4.2

Rewrite the expression.

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{6 π}{2^{2}})$

Step 15.2.2.5

Simplify.

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{6 π}{2^{2}})$

Step 15.2.3

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

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{6 π}{4})$

Step 15.2.4

Cancel the common factor of $6$ and $4$ .

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Step 15.2.4.1

Factor $2$ out of $6 π$ .

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{2 (3 π)}{4})$

Step 15.2.4.2

Cancel the common factors.

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Step 15.2.4.2.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{2 (3 π)}{2 (2)})$

Step 15.2.4.2.2

Cancel the common factor.

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{2 (3 π)}{2 \cdot 2})$

Step 15.2.4.2.3

Rewrite the expression.

$f (\frac{\sqrt{6 π}}{2}) = \sin (\frac{3 π}{2})$

Step 15.2.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (\frac{\sqrt{6 π}}{2}) = - \sin (\frac{π}{2})$

Step 15.2.6

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{\sqrt{6 π}}{2}) = - 1 \cdot 1$

Step 15.2.7

Multiply $- 1$ by $1$ .

$f (\frac{\sqrt{6 π}}{2}) = - 1$

Step 15.2.8

The final answer is $- 1$ .

$y = - 1$

Step 16

Evaluate the second derivative at $x = - \frac{\sqrt{6 π}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 4 {(- \frac{\sqrt{6 π}}{2})}^{2} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

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

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Step 17.1.1.1

Apply the product rule to $- \frac{\sqrt{6 π}}{2}$ .

$- 4 ({(- 1)}^{2} {(\frac{\sqrt{6 π}}{2})}^{2}) \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.1.2

Apply the product rule to $\frac{\sqrt{6 π}}{2}$ .

$- 4 ({(- 1)}^{2} \frac{{\sqrt{6 π}}^{2}}{2^{2}}) \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.2

Raise $- 1$ to the power of $2$ .

$- 4 (1 \frac{{\sqrt{6 π}}^{2}}{2^{2}}) \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.3

Multiply $\frac{{\sqrt{6 π}}^{2}}{2^{2}}$ by $1$ .

$- 4 \frac{{\sqrt{6 π}}^{2}}{2^{2}} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.4

Rewrite ${\sqrt{6 π}}^{2}$ as $6 π$ .

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Step 17.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 π}$ as ${(6 π)}^{\frac{1}{2}}$ .

$- 4 \frac{{({(6 π)}^{\frac{1}{2}})}^{2}}{2^{2}} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.4.2

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

$- 4 \frac{{(6 π)}^{\frac{1}{2} \cdot 2}}{2^{2}} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.4.3

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

$- 4 \frac{{(6 π)}^{\frac{2}{2}}}{2^{2}} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.4.4

Cancel the common factor of $2$ .

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Step 17.1.4.4.1

Cancel the common factor.

$- 4 \frac{{(6 π)}^{\frac{2}{2}}}{2^{2}} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.4.4.2

Rewrite the expression.

$- 4 \frac{{(6 π)}^{1}}{2^{2}} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.4.5

Simplify.

$- 4 \frac{6 π}{2^{2}} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.5

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

$- 4 \frac{6 π}{4} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.6

Cancel the common factor of $4$ .

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Step 17.1.6.1

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{6 π}{4} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.6.2

Cancel the common factor.

$4 \cdot - 1 \frac{6 π}{4} \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.6.3

Rewrite the expression.

$- 1 (6 π) \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.7

Multiply $6$ by $- 1$ .

$- 6 π \sin ({(- \frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.8

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

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Step 17.1.8.1

Apply the product rule to $- \frac{\sqrt{6 π}}{2}$ .

$- 6 π \sin ({(- 1)}^{2} {(\frac{\sqrt{6 π}}{2})}^{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.8.2

Apply the product rule to $\frac{\sqrt{6 π}}{2}$ .

$- 6 π \sin ({(- 1)}^{2} \frac{{\sqrt{6 π}}^{2}}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.9

Raise $- 1$ to the power of $2$ .

$- 6 π \sin (1 \frac{{\sqrt{6 π}}^{2}}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.10

Multiply $\frac{{\sqrt{6 π}}^{2}}{2^{2}}$ by $1$ .

$- 6 π \sin (\frac{{\sqrt{6 π}}^{2}}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.11

Rewrite ${\sqrt{6 π}}^{2}$ as $6 π$ .

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Step 17.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 π}$ as ${(6 π)}^{\frac{1}{2}}$ .

$- 6 π \sin (\frac{{({(6 π)}^{\frac{1}{2}})}^{2}}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.11.2

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

$- 6 π \sin (\frac{{(6 π)}^{\frac{1}{2} \cdot 2}}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.11.3

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

$- 6 π \sin (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.11.4

Cancel the common factor of $2$ .

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Step 17.1.11.4.1

Cancel the common factor.

$- 6 π \sin (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.11.4.2

Rewrite the expression.

$- 6 π \sin (\frac{{(6 π)}^{1}}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.11.5

Simplify.

$- 6 π \sin (\frac{6 π}{2^{2}}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.12

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

$- 6 π \sin (\frac{6 π}{4}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.13

Cancel the common factor of $6$ and $4$ .

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Step 17.1.13.1

Factor $2$ out of $6 π$ .

$- 6 π \sin (\frac{2 (3 π)}{4}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.13.2

Cancel the common factors.

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Step 17.1.13.2.1

Factor $2$ out of $4$ .

$- 6 π \sin (\frac{2 (3 π)}{2 (2)}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.13.2.2

Cancel the common factor.

$- 6 π \sin (\frac{2 (3 π)}{2 \cdot 2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.13.2.3

Rewrite the expression.

$- 6 π \sin (\frac{3 π}{2}) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.14

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- 6 π (- \sin (\frac{π}{2})) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.15

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- 6 π (- 1 \cdot 1) + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.16

Multiply $- 1$ by $1$ .

$- 6 π \cdot - 1 + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.17

Multiply $- 1$ by $- 6$ .

$6 π + 2 \cos ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.18

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

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Step 17.1.18.1

Apply the product rule to $- \frac{\sqrt{6 π}}{2}$ .

$6 π + 2 \cos ({(- 1)}^{2} {(\frac{\sqrt{6 π}}{2})}^{2})$

Step 17.1.18.2

Apply the product rule to $\frac{\sqrt{6 π}}{2}$ .

$6 π + 2 \cos ({(- 1)}^{2} \frac{{\sqrt{6 π}}^{2}}{2^{2}})$

Step 17.1.19

Raise $- 1$ to the power of $2$ .

$6 π + 2 \cos (1 \frac{{\sqrt{6 π}}^{2}}{2^{2}})$

Step 17.1.20

Multiply $\frac{{\sqrt{6 π}}^{2}}{2^{2}}$ by $1$ .

$6 π + 2 \cos (\frac{{\sqrt{6 π}}^{2}}{2^{2}})$

Step 17.1.21

Rewrite ${\sqrt{6 π}}^{2}$ as $6 π$ .

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Step 17.1.21.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 π}$ as ${(6 π)}^{\frac{1}{2}}$ .

$6 π + 2 \cos (\frac{{({(6 π)}^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 17.1.21.2

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

$6 π + 2 \cos (\frac{{(6 π)}^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 17.1.21.3

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

$6 π + 2 \cos (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}})$

Step 17.1.21.4

Cancel the common factor of $2$ .

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Step 17.1.21.4.1

Cancel the common factor.

$6 π + 2 \cos (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}})$

Step 17.1.21.4.2

Rewrite the expression.

$6 π + 2 \cos (\frac{{(6 π)}^{1}}{2^{2}})$

Step 17.1.21.5

Simplify.

$6 π + 2 \cos (\frac{6 π}{2^{2}})$

Step 17.1.22

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

$6 π + 2 \cos (\frac{6 π}{4})$

Step 17.1.23

Cancel the common factor of $6$ and $4$ .

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Step 17.1.23.1

Factor $2$ out of $6 π$ .

$6 π + 2 \cos (\frac{2 (3 π)}{4})$

Step 17.1.23.2

Cancel the common factors.

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Step 17.1.23.2.1

Factor $2$ out of $4$ .

$6 π + 2 \cos (\frac{2 (3 π)}{2 (2)})$

Step 17.1.23.2.2

Cancel the common factor.

$6 π + 2 \cos (\frac{2 (3 π)}{2 \cdot 2})$

Step 17.1.23.2.3

Rewrite the expression.

$6 π + 2 \cos (\frac{3 π}{2})$

Step 17.1.24

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$6 π + 2 \cos (\frac{π}{2})$

Step 17.1.25

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$6 π + 2 \cdot 0$

Step 17.1.26

Multiply $2$ by $0$ .

$6 π + 0$

Step 17.2

Add $6 π$ and $0$ .

$6 π$

Step 18

$x = - \frac{\sqrt{6 π}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{\sqrt{6 π}}{2}$ is a local minimum

Step 19

Find the y-value when $x = - \frac{\sqrt{6 π}}{2}$ .

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Step 19.1

Replace the variable $x$ with $- \frac{\sqrt{6 π}}{2}$ in the expression.

$f (- \frac{\sqrt{6 π}}{2}) = \sin ({(- \frac{\sqrt{6 π}}{2})}^{2})$

Step 19.2

Simplify the result.

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Step 19.2.1

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

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Step 19.2.1.1

Apply the product rule to $- \frac{\sqrt{6 π}}{2}$ .

$f (- \frac{\sqrt{6 π}}{2}) = \sin ({(- 1)}^{2} {(\frac{\sqrt{6 π}}{2})}^{2})$

Step 19.2.1.2

Apply the product rule to $\frac{\sqrt{6 π}}{2}$ .

$f (- \frac{\sqrt{6 π}}{2}) = \sin ({(- 1)}^{2} (\frac{{\sqrt{6 π}}^{2}}{2^{2}}))$

Step 19.2.2

Simplify the expression.

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Step 19.2.2.1

Raise $- 1$ to the power of $2$ .

$f (- \frac{\sqrt{6 π}}{2}) = \sin (1 (\frac{{\sqrt{6 π}}^{2}}{2^{2}}))$

Step 19.2.2.2

Multiply $\frac{{\sqrt{6 π}}^{2}}{2^{2}}$ by $1$ .

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{{\sqrt{6 π}}^{2}}{2^{2}})$

Step 19.2.3

Rewrite ${\sqrt{6 π}}^{2}$ as $6 π$ .

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Step 19.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 π}$ as ${(6 π)}^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{{({(6 π)}^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 19.2.3.2

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

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{{(6 π)}^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 19.2.3.3

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

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}})$

Step 19.2.3.4

Cancel the common factor of $2$ .

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Step 19.2.3.4.1

Cancel the common factor.

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{{(6 π)}^{\frac{2}{2}}}{2^{2}})$

Step 19.2.3.4.2

Rewrite the expression.

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{6 π}{2^{2}})$

Step 19.2.3.5

Simplify.

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{6 π}{2^{2}})$

Step 19.2.4

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

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{6 π}{4})$

Step 19.2.5

Cancel the common factor of $6$ and $4$ .

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Step 19.2.5.1

Factor $2$ out of $6 π$ .

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{2 (3 π)}{4})$

Step 19.2.5.2

Cancel the common factors.

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Step 19.2.5.2.1

Factor $2$ out of $4$ .

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{2 (3 π)}{2 (2)})$

Step 19.2.5.2.2

Cancel the common factor.

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{2 (3 π)}{2 \cdot 2})$

Step 19.2.5.2.3

Rewrite the expression.

$f (- \frac{\sqrt{6 π}}{2}) = \sin (\frac{3 π}{2})$

Step 19.2.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (- \frac{\sqrt{6 π}}{2}) = - \sin (\frac{π}{2})$

Step 19.2.7

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (- \frac{\sqrt{6 π}}{2}) = - 1 \cdot 1$

Step 19.2.8

Multiply $- 1$ by $1$ .

$f (- \frac{\sqrt{6 π}}{2}) = - 1$

Step 19.2.9

The final answer is $- 1$ .

$y = - 1$

Step 20

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

$(0, 0)$ is a local minima

$(\frac{\sqrt{6 π}}{2}, - 1)$ is a local minima

$(- \frac{\sqrt{6 π}}{2}, - 1)$ is a local minima

Step 21