Calculus Examples

$f (x) = \sqrt{3} \sin (x) + \cos (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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

$\frac{d}{d x} [\sqrt{3} \sin (x)] + \frac{d}{d x} [\cos (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [\sqrt{3} \sin (x)]$ .

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Step 1.2.1

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

$\sqrt{3} \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)]$

Step 1.2.2

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

$\sqrt{3} \cos (x) + \frac{d}{d x} [\cos (x)]$

Step 1.3

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

$\sqrt{3} \cos (x) - \sin (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

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

$f'' (x) = \frac{d}{d x} (\sqrt{3} \cos (x)) + \frac{d}{d x} (- \sin (x))$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = \sqrt{3} \frac{d}{d x} (\cos (x)) + \frac{d}{d x} (- \sin (x))$

Step 2.2.2

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

$f'' (x) = \sqrt{3} (- \sin (x)) + \frac{d}{d x} (- \sin (x))$

Step 2.3

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

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Step 2.3.1

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

$f'' (x) = \sqrt{3} (- \sin (x)) - \frac{d}{d x} \sin (x)$

Step 2.3.2

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

$f'' (x) = \sqrt{3} (- \sin (x)) - \cos (x)$

Step 2.4

Reorder terms.

$f'' (x) = - \sqrt{3} \sin (x) - \cos (x)$

Step 3

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

$\sqrt{3} \cos (x) - \sin (x) = 0$

Step 4

Divide each term in the equation by $\cos (x)$ .

$\frac{\sqrt{3} \cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5

Cancel the common factor of $\cos (x)$ .

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Step 5.1

Cancel the common factor.

$\frac{\sqrt{3} \cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5.2

Divide $\sqrt{3}$ by $1$ .

$\sqrt{3} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6

Separate fractions.

$\sqrt{3} + \frac{- 1}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\sqrt{3} + \frac{- 1}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 8

Divide $- 1$ by $1$ .

$\sqrt{3} - \tan (x) = \frac{0}{\cos (x)}$

Step 9

Separate fractions.

$\sqrt{3} - \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 10

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sqrt{3} - \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 11

Divide $0$ by $1$ .

$\sqrt{3} - \tan (x) = 0 \sec (x)$

Step 12

Multiply $0$ by $\sec (x)$ .

$\sqrt{3} - \tan (x) = 0$

Step 13

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

$- \tan (x) = - \sqrt{3}$

Step 14

Divide each term in $- \tan (x) = - \sqrt{3}$ by $- 1$ and simplify.

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Step 14.1

Divide each term in $- \tan (x) = - \sqrt{3}$ by $- 1$ .

$\frac{- \tan (x)}{- 1} = \frac{- \sqrt{3}}{- 1}$

Step 14.2

Simplify the left side.

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Step 14.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (x)}{1} = \frac{- \sqrt{3}}{- 1}$

Step 14.2.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{- \sqrt{3}}{- 1}$

Step 14.3

Simplify the right side.

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Step 14.3.1

Dividing two negative values results in a positive value.

$\tan (x) = \frac{\sqrt{3}}{1}$

Step 14.3.2

Divide $\sqrt{3}$ by $1$ .

$\tan (x) = \sqrt{3}$

Step 15

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

$x = \arctan (\sqrt{3})$

Step 16

Simplify the right side.

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Step 16.1

The exact value of $\arctan (\sqrt{3})$ is $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 17

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{3}$

Step 18

Simplify $π + \frac{π}{3}$ .

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Step 18.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = π \cdot \frac{3}{3} + \frac{π}{3}$

Step 18.2

Combine fractions.

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Step 18.2.1

Combine $π$ and $\frac{3}{3}$ .

$x = \frac{π \cdot 3}{3} + \frac{π}{3}$

Step 18.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 + π}{3}$

Step 18.3

Simplify the numerator.

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Step 18.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π + π}{3}$

Step 18.3.2

Add $3 π$ and $π$ .

$x = \frac{4 π}{3}$

Step 19

The solution to the equation $x = \frac{π}{3}$ .

$x = \frac{π}{3}, \frac{4 π}{3}$

Step 20

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

$- \sqrt{3} \sin (\frac{π}{3}) - \cos (\frac{π}{3})$

Step 21

Evaluate the second derivative.

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Step 21.1

Simplify each term.

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Step 21.1.1

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$- \sqrt{3} \frac{\sqrt{3}}{2} - \cos (\frac{π}{3})$

Step 21.1.2

Multiply $- \sqrt{3} \frac{\sqrt{3}}{2}$ .

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Step 21.1.2.1

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

$- \frac{\sqrt{3} \sqrt{3}}{2} - \cos (\frac{π}{3})$

Step 21.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{{\sqrt{3}}^{1} \sqrt{3}}{2} - \cos (\frac{π}{3})$

Step 21.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2} - \cos (\frac{π}{3})$

Step 21.1.2.4

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

$- \frac{{\sqrt{3}}^{1 + 1}}{2} - \cos (\frac{π}{3})$

Step 21.1.2.5

Add $1$ and $1$ .

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

Step 21.1.3

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

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Step 21.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{{(3^{\frac{1}{2}})}^{2}}{2} - \cos (\frac{π}{3})$

Step 21.1.3.2

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

$- \frac{3^{\frac{1}{2} \cdot 2}}{2} - \cos (\frac{π}{3})$

Step 21.1.3.3

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

$- \frac{3^{\frac{2}{2}}}{2} - \cos (\frac{π}{3})$

Step 21.1.3.4

Cancel the common factor of $2$ .

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Step 21.1.3.4.1

Cancel the common factor.

$- \frac{3^{\frac{2}{2}}}{2} - \cos (\frac{π}{3})$

Step 21.1.3.4.2

Rewrite the expression.

$- \frac{3^{1}}{2} - \cos (\frac{π}{3})$

Step 21.1.3.5

Evaluate the exponent.

$- \frac{3}{2} - \cos (\frac{π}{3})$

Step 21.1.4

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- \frac{3}{2} - \frac{1}{2}$

Step 21.2

Combine fractions.

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Step 21.2.1

Combine the numerators over the common denominator.

$\frac{- 3 - 1}{2}$

Step 21.2.2

Simplify the expression.

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Step 21.2.2.1

Subtract $1$ from $- 3$ .

$\frac{- 4}{2}$

Step 21.2.2.2

Divide $- 4$ by $2$ .

$- 2$

Step 22

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

$x = \frac{π}{3}$ is a local maximum

Step 23

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

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Step 23.1

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

$f (\frac{π}{3}) = \sqrt{3} \sin (\frac{π}{3}) + \cos (\frac{π}{3})$

Step 23.2

Simplify the result.

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Step 23.2.1

Simplify each term.

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Step 23.2.1.1

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$f (\frac{π}{3}) = \sqrt{3} (\frac{\sqrt{3}}{2}) + \cos (\frac{π}{3})$

Step 23.2.1.2

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

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Step 23.2.1.2.1

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

$f (\frac{π}{3}) = \frac{\sqrt{3} \sqrt{3}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{π}{3}) = \frac{\sqrt{3} \sqrt{3}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{π}{3}) = \frac{\sqrt{3} \sqrt{3}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.2.4

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

$f (\frac{π}{3}) = \frac{{\sqrt{3}}^{1 + 1}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.2.5

Add $1$ and $1$ .

$f (\frac{π}{3}) = \frac{{\sqrt{3}}^{2}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.3

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

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Step 23.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{π}{3}) = \frac{{(3^{\frac{1}{2}})}^{2}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.3.2

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

$f (\frac{π}{3}) = \frac{3^{\frac{1}{2} \cdot 2}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.3.3

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

$f (\frac{π}{3}) = \frac{3^{\frac{2}{2}}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.3.4

Cancel the common factor of $2$ .

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Step 23.2.1.3.4.1

Cancel the common factor.

$f (\frac{π}{3}) = \frac{3^{\frac{2}{2}}}{2} + \cos (\frac{π}{3})$

Step 23.2.1.3.4.2

Rewrite the expression.

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

Step 23.2.1.3.5

Evaluate the exponent.

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

Step 23.2.1.4

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$f (\frac{π}{3}) = \frac{3}{2} + \frac{1}{2}$

Step 23.2.2

Combine fractions.

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Step 23.2.2.1

Combine the numerators over the common denominator.

$f (\frac{π}{3}) = \frac{3 + 1}{2}$

Step 23.2.2.2

Simplify the expression.

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Step 23.2.2.2.1

Add $3$ and $1$ .

$f (\frac{π}{3}) = \frac{4}{2}$

Step 23.2.2.2.2

Divide $4$ by $2$ .

$f (\frac{π}{3}) = 2$

Step 23.2.3

The final answer is $2$ .

$y = 2$

Step 24

Evaluate the second derivative at $x = \frac{4 π}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \sqrt{3} \sin (\frac{4 π}{3}) - \cos (\frac{4 π}{3})$

Step 25

Evaluate the second derivative.

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Step 25.1

Simplify each term.

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Step 25.1.1

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 third quadrant.

$- \sqrt{3} (- \sin (\frac{π}{3})) - \cos (\frac{4 π}{3})$

Step 25.1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 25.1.3

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

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Step 25.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt{3} \frac{\sqrt{3}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.3.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 25.1.3.3

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

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

Step 25.1.3.4

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} \sqrt{3}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.3.5

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.3.6

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

$\frac{{\sqrt{3}}^{1 + 1}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.3.7

Add $1$ and $1$ .

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

Step 25.1.4

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

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Step 25.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.4.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.4.3

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

$\frac{3^{\frac{2}{2}}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.4.4

Cancel the common factor of $2$ .

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Step 25.1.4.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.4.4.2

Rewrite the expression.

$\frac{3^{1}}{2} - \cos (\frac{4 π}{3})$

Step 25.1.4.5

Evaluate the exponent.

$\frac{3}{2} - \cos (\frac{4 π}{3})$

Step 25.1.5

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

$\frac{3}{2} - - \cos (\frac{π}{3})$

Step 25.1.6

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{3}{2} - - \frac{1}{2}$

Step 25.1.7

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

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Step 25.1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 25.1.7.2

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

$\frac{3}{2} + \frac{1}{2}$

Step 25.2

Combine fractions.

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Step 25.2.1

Combine the numerators over the common denominator.

$\frac{3 + 1}{2}$

Step 25.2.2

Simplify the expression.

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Step 25.2.2.1

Add $3$ and $1$ .

$\frac{4}{2}$

Step 25.2.2.2

Divide $4$ by $2$ .

$2$

Step 26

$x = \frac{4 π}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{4 π}{3}$ is a local minimum

Step 27

Find the y-value when $x = \frac{4 π}{3}$ .

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Step 27.1

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

$f (\frac{4 π}{3}) = \sqrt{3} \sin (\frac{4 π}{3}) + \cos (\frac{4 π}{3})$

Step 27.2

Simplify the result.

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Step 27.2.1

Simplify each term.

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Step 27.2.1.1

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 third quadrant.

$f (\frac{4 π}{3}) = \sqrt{3} (- \sin (\frac{π}{3})) + \cos (\frac{4 π}{3})$

Step 27.2.1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$f (\frac{4 π}{3}) = \sqrt{3} (- \frac{\sqrt{3}}{2}) + \cos (\frac{4 π}{3})$

Step 27.2.1.3

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

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Step 27.2.1.3.1

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

$f (\frac{4 π}{3}) = - \frac{\sqrt{3} \sqrt{3}}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{4 π}{3}) = - \frac{\sqrt{3} \sqrt{3}}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{4 π}{3}) = - \frac{\sqrt{3} \sqrt{3}}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.3.4

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

$f (\frac{4 π}{3}) = - \frac{{\sqrt{3}}^{1 + 1}}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.3.5

Add $1$ and $1$ .

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

Step 27.2.1.4

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

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Step 27.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{4 π}{3}) = - \frac{{(3^{\frac{1}{2}})}^{2}}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.4.2

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

$f (\frac{4 π}{3}) = - \frac{3^{\frac{1}{2} \cdot 2}}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.4.3

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

$f (\frac{4 π}{3}) = - \frac{3^{\frac{2}{2}}}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.4.4

Cancel the common factor of $2$ .

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Step 27.2.1.4.4.1

Cancel the common factor.

$f (\frac{4 π}{3}) = - \frac{3^{\frac{2}{2}}}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.4.4.2

Rewrite the expression.

$f (\frac{4 π}{3}) = - \frac{3}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.4.5

Evaluate the exponent.

$f (\frac{4 π}{3}) = - \frac{3}{2} + \cos (\frac{4 π}{3})$

Step 27.2.1.5

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

$f (\frac{4 π}{3}) = - \frac{3}{2} - \cos (\frac{π}{3})$

Step 27.2.1.6

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$f (\frac{4 π}{3}) = - \frac{3}{2} - \frac{1}{2}$

Step 27.2.2

Combine fractions.

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Step 27.2.2.1

Combine the numerators over the common denominator.

$f (\frac{4 π}{3}) = \frac{- 3 - 1}{2}$

Step 27.2.2.2

Simplify the expression.

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Step 27.2.2.2.1

Subtract $1$ from $- 3$ .

$f (\frac{4 π}{3}) = \frac{- 4}{2}$

Step 27.2.2.2.2

Divide $- 4$ by $2$ .

$f (\frac{4 π}{3}) = - 2$

Step 27.2.3

The final answer is $- 2$ .

$y = - 2$

Step 28

These are the local extrema for $f (x) = \sqrt{3} \sin (x) + \cos (x)$ .

$(\frac{π}{3}, 2)$ is a local maxima

$(\frac{4 π}{3}, - 2)$ is a local minima

Step 29