Calculus Examples

$3 \cos^{2} (x) - 6 \sin (x)$

Step 1

Write $3 \cos^{2} (x) - 6 \sin (x)$ as a function.

$f (x) = 3 \cos^{2} (x) - 6 \sin (x)$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

By the Sum Rule, the derivative of $3 \cos^{2} (x) - 6 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [3 \cos^{2} (x)] + \frac{d}{d x} [- 6 \sin (x)]$ .

$\frac{d}{d x} [3 \cos^{2} (x)] + \frac{d}{d x} [- 6 \sin (x)]$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

$3 \frac{d}{d x} [\cos^{2} (x)] + \frac{d}{d x} [- 6 \sin (x)]$

Step 2.1.1.2.2

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

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Step 2.1.1.2.2.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

$3 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \sin (x)]$

Step 2.1.1.2.2.2

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

$3 (2 u \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \sin (x)]$

Step 2.1.1.2.2.3

Replace all occurrences of $u$ with $\cos (x)$ .

$3 (2 \cos (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \sin (x)]$

Step 2.1.1.2.3

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

$3 (2 \cos (x) (- \sin (x))) + \frac{d}{d x} [- 6 \sin (x)]$

Step 2.1.1.2.4

Multiply $- 1$ by $2$ .

$3 (- 2 \cos (x) \sin (x)) + \frac{d}{d x} [- 6 \sin (x)]$

Step 2.1.1.2.5

Multiply $- 2$ by $3$ .

$- 6 \cos (x) \sin (x) + \frac{d}{d x} [- 6 \sin (x)]$

Step 2.1.1.3

Evaluate $\frac{d}{d x} [- 6 \sin (x)]$ .

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Step 2.1.1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \sin (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\sin (x)]$ .

$- 6 \cos (x) \sin (x) - 6 \frac{d}{d x} [\sin (x)]$

Step 2.1.1.3.2

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

$f' (x) = - 6 \cos (x) \sin (x) - 6 \cos (x)$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $- 6 \cos (x) \sin (x) - 6 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [- 6 \cos (x) \sin (x)] + \frac{d}{d x} [- 6 \cos (x)]$ .

$\frac{d}{d x} [- 6 \cos (x) \sin (x)] + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2

Evaluate $\frac{d}{d x} [- 6 \cos (x) \sin (x)]$ .

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Step 2.1.2.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \cos (x) \sin (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\cos (x) \sin (x)]$ .

$- 6 \frac{d}{d x} [\cos (x) \sin (x)] + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.2

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 $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

$- 6 (\cos (x) \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.3

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

$- 6 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.4

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

$- 6 (\cos (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.5

Raise $\cos (x)$ to the power of $1$ .

$- 6 (\cos^{1} (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.6

Raise $\cos (x)$ to the power of $1$ .

$- 6 (\cos^{1} (x) \cos^{1} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.7

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

$- 6 ({\cos (x)}^{1 + 1} + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.8

Add $1$ and $1$ .

$- 6 (\cos^{2} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.9

Raise $\sin (x)$ to the power of $1$ .

$- 6 (\cos^{2} (x) - (\sin^{1} (x) \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.10

Raise $\sin (x)$ to the power of $1$ .

$- 6 (\cos^{2} (x) - (\sin^{1} (x) \sin^{1} (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.11

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

$- 6 (\cos^{2} (x) - {\sin (x)}^{1 + 1}) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.2.12

Add $1$ and $1$ .

$- 6 (\cos^{2} (x) - \sin^{2} (x)) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [- 6 \cos (x)]$ .

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Step 2.1.2.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \cos (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\cos (x)]$ .

$- 6 (\cos^{2} (x) - \sin^{2} (x)) - 6 \frac{d}{d x} [\cos (x)]$

Step 2.1.2.3.2

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

$- 6 (\cos^{2} (x) - \sin^{2} (x)) - 6 (- \sin (x))$

Step 2.1.2.3.3

Multiply $- 1$ by $- 6$ .

$- 6 (\cos^{2} (x) - \sin^{2} (x)) + 6 \sin (x)$

Step 2.1.2.4

Simplify.

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Step 2.1.2.4.1

Apply the distributive property.

$- 6 \cos^{2} (x) - 6 (- \sin^{2} (x)) + 6 \sin (x)$

Step 2.1.2.4.2

Multiply $- 1$ by $- 6$ .

$f'' (x) = - 6 \cos^{2} (x) + 6 \sin^{2} (x) + 6 \sin (x)$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 6 \cos^{2} (x) + 6 \sin^{2} (x) + 6 \sin (x)$ .

$- 6 \cos^{2} (x) + 6 \sin^{2} (x) + 6 \sin (x)$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $- 6 \cos^{2} (x) + 6 \sin^{2} (x) + 6 \sin (x) = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$- 6 \cos^{2} (x) + 6 \sin^{2} (x) + 6 \sin (x) = 0$

Step 2.2.2

Replace the $- 6 \cos^{2} (x)$ with $- 6 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$- 6 (1 - \sin^{2} (x)) + 6 \sin^{2} (x) + 6 \sin (x) = 0$

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

Apply the distributive property.

$- 6 \cdot 1 - 6 (- \sin^{2} (x)) + 6 \sin^{2} (x) + 6 \sin (x) = 0$

Step 2.2.3.2

Multiply $- 6$ by $1$ .

$- 6 - 6 (- \sin^{2} (x)) + 6 \sin^{2} (x) + 6 \sin (x) = 0$

Step 2.2.3.3

Multiply $- 1$ by $- 6$ .

$- 6 + 6 \sin^{2} (x) + 6 \sin^{2} (x) + 6 \sin (x) = 0$

Step 2.2.4

Add $6 \sin^{2} (x)$ and $6 \sin^{2} (x)$ .

$- 6 + 12 \sin^{2} (x) + 6 \sin (x) = 0$

Step 2.2.5

Reorder the polynomial.

$12 \sin^{2} (x) + 6 \sin (x) - 6 = 0$

Step 2.2.6

Substitute $u$ for $\sin (x)$ .

$12 {(u)}^{2} + 6 (u) - 6 = 0$

Step 2.2.7

Factor the left side of the equation.

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Step 2.2.7.1

Factor $6$ out of $12 u^{2} + 6 u - 6$ .

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Step 2.2.7.1.1

Factor $6$ out of $12 u^{2}$ .

$6 (2 u^{2}) + 6 u - 6 = 0$

Step 2.2.7.1.2

Factor $6$ out of $6 u$ .

$6 (2 u^{2}) + 6 u - 6 = 0$

Step 2.2.7.1.3

Factor $6$ out of $- 6$ .

$6 (2 u^{2}) + 6 u + 6 (- 1) = 0$

Step 2.2.7.1.4

Factor $6$ out of $6 (2 u^{2}) + 6 u$ .

$6 (2 u^{2} + u) + 6 (- 1) = 0$

Step 2.2.7.1.5

Factor $6$ out of $6 (2 u^{2} + u) + 6 (- 1)$ .

$6 (2 u^{2} + u - 1) = 0$

Step 2.2.7.2

Factor.

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Step 2.2.7.2.1

Factor by grouping.

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Step 2.2.7.2.1.1

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

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Step 2.2.7.2.1.1.1

Multiply by $1$ .

$6 (2 u^{2} + 1 u - 1) = 0$

Step 2.2.7.2.1.1.2

Rewrite $1$ as $- 1$ plus $2$

$6 (2 u^{2} + (- 1 + 2) u - 1) = 0$

Step 2.2.7.2.1.1.3

Apply the distributive property.

$6 (2 u^{2} - 1 u + 2 u - 1) = 0$

Step 2.2.7.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.7.2.1.2.1

Group the first two terms and the last two terms.

$6 (2 u^{2} - 1 u + 2 u - 1) = 0$

Step 2.2.7.2.1.2.2

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

$6 (u (2 u - 1) + 1 (2 u - 1)) = 0$

Step 2.2.7.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$6 ((2 u - 1) (u + 1)) = 0$

Step 2.2.7.2.2

Remove unnecessary parentheses.

$6 (2 u - 1) (u + 1) = 0$

Step 2.2.8

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

$2 u - 1 = 0$

$u + 1 = 0$

Step 2.2.9

Set $2 u - 1$ equal to $0$ and solve for $u$ .

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Step 2.2.9.1

Set $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 2.2.9.2

Solve $2 u - 1 = 0$ for $u$ .

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Step 2.2.9.2.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 2.2.9.2.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 2.2.9.2.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 2.2.9.2.2.2

Simplify the left side.

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Step 2.2.9.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.9.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 2.2.9.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 2.2.10

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 2.2.10.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 2.2.10.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 2.2.11

The final solution is all the values that make $6 (2 u - 1) (u + 1) = 0$ true.

$u = \frac{1}{2}, - 1$

Step 2.2.12

Substitute $\sin (x)$ for $u$ .

$\sin (x) = \frac{1}{2}, - 1$

Step 2.2.13

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{1}{2}$

$\sin (x) = - 1$

Step 2.2.14

Solve for $x$ in $\sin (x) = \frac{1}{2}$ .

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Step 2.2.14.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (\frac{1}{2})$

Step 2.2.14.2

Simplify the right side.

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Step 2.2.14.2.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 2.2.14.3

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

$x = π - \frac{π}{6}$

Step 2.2.14.4

Simplify $π - \frac{π}{6}$ .

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Step 2.2.14.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 2.2.14.4.2

Combine fractions.

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Step 2.2.14.4.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 2.2.14.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 2.2.14.4.3

Simplify the numerator.

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Step 2.2.14.4.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 2.2.14.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 2.2.14.5

Find the period of $\sin (x)$ .

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Step 2.2.14.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.2.14.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.2.14.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.2.14.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.2.14.6

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 2.2.15

Solve for $x$ in $\sin (x) = - 1$ .

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Step 2.2.15.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (- 1)$

Step 2.2.15.2

Simplify the right side.

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Step 2.2.15.2.1

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

$x = - \frac{π}{2}$

Step 2.2.15.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{2} + π$

Step 2.2.15.4

Simplify the expression to find the second solution.

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Step 2.2.15.4.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 2.2.15.4.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

Step 2.2.15.5

Find the period of $\sin (x)$ .

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Step 2.2.15.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.2.15.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.2.15.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.2.15.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.2.15.6

Add $2 π$ to every negative angle to get positive angles.

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Step 2.2.15.6.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 2.2.15.6.2

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 2.2.15.6.3

Combine fractions.

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Step 2.2.15.6.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 2.2.15.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 2.2.15.6.4

Simplify the numerator.

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Step 2.2.15.6.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 2.2.15.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 2.2.15.6.5

List the new angles.

$x = \frac{3 π}{2}$

Step 2.2.15.7

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 2.2.16

List all of the solutions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 2.2.17

Consolidate the answers.

$x = \frac{π}{6} + \frac{2 π n}{3}$ , for any integer $n$

Step 3

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, \infty)$

Step 5

Substitute any number from the interval $(- \infty, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (0) = - 6 \cos^{2} (0) + 6 \sin^{2} (0) + 6 \sin (0)$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f'' (0) = - 6 \cdot 1^{2} + 6 \sin^{2} (0) + 6 \sin (0)$

Step 5.2.1.2

One to any power is one.

$f'' (0) = - 6 \cdot 1 + 6 \sin^{2} (0) + 6 \sin (0)$

Step 5.2.1.3

Multiply $- 6$ by $1$ .

$f'' (0) = - 6 + 6 \sin^{2} (0) + 6 \sin (0)$

Step 5.2.1.4

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

$f'' (0) = - 6 + 6 \cdot 0^{2} + 6 \sin (0)$

Step 5.2.1.5

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

$f'' (0) = - 6 + 6 \cdot 0 + 6 \sin (0)$

Step 5.2.1.6

Multiply $6$ by $0$ .

$f'' (0) = - 6 + 0 + 6 \sin (0)$

Step 5.2.1.7

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

$f'' (0) = - 6 + 0 + 6 \cdot 0$

Step 5.2.1.8

Multiply $6$ by $0$ .

$f'' (0) = - 6 + 0 + 0$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 6$ and $0$ .

$f'' (0) = - 6 + 0$

Step 5.2.2.2

Add $- 6$ and $0$ .

$f'' (0) = - 6$

Step 5.2.3

The final answer is $- 6$ .

$- 6$

Step 5.3

The graph is concave down on the interval $(- \infty, \infty)$ because $f'' (0)$ is negative.

Concave down on $(- \infty, \infty)$ since $f'' (x)$ is negative

Step 6