Calculus Examples

$f (x) = 2 x - \tan (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

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

$2 - \frac{d}{d x} [\tan (x)]$

Step 1.3.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$2 - \sec^{2} (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

$f'' (x) = 0 + \frac{d}{d x} (- \sec^{2} (x))$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = 0 - \frac{d}{d x} \sec^{2} (x)$

Step 2.2.2

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 $g (x) = \sec (x)$ .

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Step 2.2.2.1

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

$f'' (x) = 0 - (\frac{d}{d u} (u^{2}) \frac{d}{d x} (\sec (x)))$

Step 2.2.2.2

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

$f'' (x) = 0 - (2 u \frac{d}{d x} (\sec (x)))$

Step 2.2.2.3

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

$f'' (x) = 0 - (2 \sec (x) \frac{d}{d x} (\sec (x)))$

Step 2.2.3

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$f'' (x) = 0 - (2 \sec (x) (\sec (x) \tan (x)))$

Step 2.2.4

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

$f'' (x) = 0 - (2 (\sec (x) \sec (x)) \tan (x))$

Step 2.2.5

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

$f'' (x) = 0 - (2 (\sec (x) \sec (x)) \tan (x))$

Step 2.2.6

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

$f'' (x) = 0 - (2 {\sec (x)}^{1 + 1} \tan (x))$

Step 2.2.7

Add $1$ and $1$ .

$f'' (x) = 0 - (2 \sec^{2} (x) \tan (x))$

Step 2.2.8

Multiply $2$ by $- 1$ .

$f'' (x) = 0 - 2 \sec^{2} (x) \tan (x)$

Step 2.3

Subtract $2 \sec^{2} (x) \tan (x)$ from $0$ .

$f'' (x) = - 2 \sec^{2} (x) \tan (x)$

Step 3

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

$2 - \sec^{2} (x) = 0$

Step 4

Subtract $2$ from both sides of the equation.

$- \sec^{2} (x) = - 2$

Step 5

Divide each term in $- \sec^{2} (x) = - 2$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \sec^{2} (x) = - 2$ by $- 1$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2

Divide $\sec^{2} (x)$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $- 2$ by $- 1$ .

$\sec^{2} (x) = 2$

Step 6

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

$\sec (x) = \pm \sqrt{2}$

Step 7

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

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Step 7.1

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

$\sec (x) = \sqrt{2}$

Step 7.2

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

$\sec (x) = - \sqrt{2}$

Step 7.3

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

$\sec (x) = \sqrt{2}, - \sqrt{2}$

Step 8

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

$\sec (x) = \sqrt{2}$

$\sec (x) = - \sqrt{2}$

Step 9

Solve for $x$ in $\sec (x) = \sqrt{2}$ .

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Step 9.1

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

$x = arcsec (\sqrt{2})$

Step 9.2

Simplify the right side.

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Step 9.2.1

The exact value of $arcsec (\sqrt{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 9.3

The secant 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.

$x = 2 π - \frac{π}{4}$

Step 9.4

Simplify $2 π - \frac{π}{4}$ .

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Step 9.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 9.4.2

Combine fractions.

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Step 9.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 9.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 9.4.3

Simplify the numerator.

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Step 9.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 9.4.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 9.5

The solution to the equation $x = \frac{π}{4}$ .

$x = \frac{π}{4}, \frac{7 π}{4}$

Step 10

Solve for $x$ in $\sec (x) = - \sqrt{2}$ .

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Step 10.1

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

$x = arcsec (- \sqrt{2})$

Step 10.2

Simplify the right side.

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Step 10.2.1

The exact value of $arcsec (- \sqrt{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 10.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{3 π}{4}$

Step 10.4

Simplify $2 π - \frac{3 π}{4}$ .

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Step 10.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 10.4.2

Combine fractions.

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Step 10.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 10.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 10.4.3

Simplify the numerator.

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Step 10.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 10.4.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 10.5

The solution to the equation $x = \frac{3 π}{4}$ .

$x = \frac{3 π}{4}, \frac{5 π}{4}$

Step 11

List all of the solutions.

$x = \frac{π}{4}, \frac{7 π}{4}, \frac{3 π}{4}, \frac{5 π}{4}$

Step 12

Exclude the solutions that do not make $2 - \sec^{2} (x) = 0$ true.

$x = \frac{π}{4}, \frac{7 π}{4}, \frac{3 π}{4}, \frac{5 π}{4}$

Step 13

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

$- 2 \sec^{2} (\frac{π}{4}) \tan (\frac{π}{4})$

Step 14

Evaluate the second derivative.

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Step 14.1

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

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

Step 14.2

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

$- 2 {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2} \tan (\frac{π}{4})$

Step 14.3

Combine and simplify the denominator.

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Step 14.3.1

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

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

Step 14.3.2

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

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

Step 14.3.3

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

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

Step 14.3.4

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

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

Step 14.3.5

Add $1$ and $1$ .

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

Step 14.3.6

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

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Step 14.3.6.1

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

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

Step 14.3.6.2

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

$- 2 {(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2} \tan (\frac{π}{4})$

Step 14.3.6.3

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

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

Step 14.3.6.4

Cancel the common factor of $2$ .

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Step 14.3.6.4.1

Cancel the common factor.

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

Step 14.3.6.4.2

Rewrite the expression.

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

Step 14.3.6.5

Evaluate the exponent.

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

Step 14.4

Cancel the common factor of $2$ .

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Step 14.4.1

Cancel the common factor.

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

Step 14.4.2

Divide $\sqrt{2}$ by $1$ .

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

Step 14.5

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

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Step 14.5.1

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

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

Step 14.5.2

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

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

Step 14.5.3

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

$- 2 \cdot 2^{\frac{2}{2}} \tan (\frac{π}{4})$

Step 14.5.4

Cancel the common factor of $2$ .

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Step 14.5.4.1

Cancel the common factor.

$- 2 \cdot 2^{\frac{2}{2}} \tan (\frac{π}{4})$

Step 14.5.4.2

Rewrite the expression.

$- 2 \cdot 2^{1} \tan (\frac{π}{4})$

Step 14.5.5

Evaluate the exponent.

$- 2 \cdot 2 \tan (\frac{π}{4})$

Step 14.6

Multiply $- 2$ by $2$ .

$- 4 \tan (\frac{π}{4})$

Step 14.7

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$- 4 \cdot 1$

Step 14.8

Multiply $- 4$ by $1$ .

$- 4$

Step 15

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

$x = \frac{π}{4}$ is a local maximum

Step 16

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

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Step 16.1

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

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

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

Cancel the common factor of $2$ .

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Step 16.2.1.1.1

Factor $2$ out of $4$ .

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

Step 16.2.1.1.2

Cancel the common factor.

$f (\frac{π}{4}) = 2 (\frac{π}{2 \cdot 2}) - \tan (\frac{π}{4})$

Step 16.2.1.1.3

Rewrite the expression.

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

Step 16.2.1.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

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

Step 16.2.1.3

Multiply $- 1$ by $1$ .

$f (\frac{π}{4}) = \frac{π}{2} - 1$

Step 16.2.2

The final answer is $\frac{π}{2} - 1$ .

$y = \frac{π}{2} - 1$

Step 17

Evaluate the second derivative at $x = \frac{7 π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2 \sec^{2} (\frac{7 π}{4}) \tan (\frac{7 π}{4})$

Step 18

Evaluate the second derivative.

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Step 18.1

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

$- 2 \sec^{2} (\frac{π}{4}) \tan (\frac{7 π}{4})$

Step 18.2

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$- 2 {(\frac{2}{\sqrt{2}})}^{2} \tan (\frac{7 π}{4})$

Step 18.3

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

$- 2 {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4

Combine and simplify the denominator.

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Step 18.4.1

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

$- 2 {(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.2

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

$- 2 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.3

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

$- 2 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.4

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

$- 2 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.5

Add $1$ and $1$ .

$- 2 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.6

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

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Step 18.4.6.1

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

$- 2 {(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.6.2

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

$- 2 {(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.6.3

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

$- 2 {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.6.4

Cancel the common factor of $2$ .

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Step 18.4.6.4.1

Cancel the common factor.

$- 2 {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.6.4.2

Rewrite the expression.

$- 2 {(\frac{2 \sqrt{2}}{2^{1}})}^{2} \tan (\frac{7 π}{4})$

Step 18.4.6.5

Evaluate the exponent.

$- 2 {(\frac{2 \sqrt{2}}{2})}^{2} \tan (\frac{7 π}{4})$

Step 18.5

Cancel the common factor of $2$ .

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Step 18.5.1

Cancel the common factor.

$- 2 {(\frac{2 \sqrt{2}}{2})}^{2} \tan (\frac{7 π}{4})$

Step 18.5.2

Divide $\sqrt{2}$ by $1$ .

$- 2 {\sqrt{2}}^{2} \tan (\frac{7 π}{4})$

Step 18.6

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

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Step 18.6.1

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

$- 2 {(2^{\frac{1}{2}})}^{2} \tan (\frac{7 π}{4})$

Step 18.6.2

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

$- 2 \cdot 2^{\frac{1}{2} \cdot 2} \tan (\frac{7 π}{4})$

Step 18.6.3

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

$- 2 \cdot 2^{\frac{2}{2}} \tan (\frac{7 π}{4})$

Step 18.6.4

Cancel the common factor of $2$ .

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Step 18.6.4.1

Cancel the common factor.

$- 2 \cdot 2^{\frac{2}{2}} \tan (\frac{7 π}{4})$

Step 18.6.4.2

Rewrite the expression.

$- 2 \cdot 2^{1} \tan (\frac{7 π}{4})$

Step 18.6.5

Evaluate the exponent.

$- 2 \cdot 2 \tan (\frac{7 π}{4})$

Step 18.7

Multiply $- 2$ by $2$ .

$- 4 \tan (\frac{7 π}{4})$

Step 18.8

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

$- 4 (- \tan (\frac{π}{4}))$

Step 18.9

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$- 4 (- 1 \cdot 1)$

Step 18.10

Multiply $- 4 (- 1 \cdot 1)$ .

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Step 18.10.1

Multiply $- 1$ by $1$ .

$- 4 \cdot - 1$

Step 18.10.2

Multiply $- 4$ by $- 1$ .

$4$

Step 19

$x = \frac{7 π}{4}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{7 π}{4}$ is a local minimum

Step 20

Find the y-value when $x = \frac{7 π}{4}$ .

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Step 20.1

Replace the variable $x$ with $\frac{7 π}{4}$ in the expression.

$f (\frac{7 π}{4}) = 2 (\frac{7 π}{4}) - \tan (\frac{7 π}{4})$

Step 20.2

Simplify the result.

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Step 20.2.1

Simplify each term.

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Step 20.2.1.1

Cancel the common factor of $2$ .

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Step 20.2.1.1.1

Factor $2$ out of $4$ .

$f (\frac{7 π}{4}) = 2 (\frac{7 π}{2 (2)}) - \tan (\frac{7 π}{4})$

Step 20.2.1.1.2

Cancel the common factor.

$f (\frac{7 π}{4}) = 2 (\frac{7 π}{2 \cdot 2}) - \tan (\frac{7 π}{4})$

Step 20.2.1.1.3

Rewrite the expression.

$f (\frac{7 π}{4}) = \frac{7 π}{2} - \tan (\frac{7 π}{4})$

Step 20.2.1.2

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

$f (\frac{7 π}{4}) = \frac{7 π}{2} + \tan (\frac{π}{4})$

Step 20.2.1.3

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$f (\frac{7 π}{4}) = \frac{7 π}{2} - (- 1 \cdot 1)$

Step 20.2.1.4

Multiply $- (- 1 \cdot 1)$ .

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Step 20.2.1.4.1

Multiply $- 1$ by $1$ .

$f (\frac{7 π}{4}) = \frac{7 π}{2} + 1$

Step 20.2.1.4.2

Multiply $- 1$ by $- 1$ .

$f (\frac{7 π}{4}) = \frac{7 π}{2} + 1$

Step 20.2.2

The final answer is $\frac{7 π}{2} + 1$ .

$y = \frac{7 π}{2} + 1$

Step 21

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

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

Step 22

Evaluate the second derivative.

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Step 22.1

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

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

Step 22.2

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

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

Step 22.3

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

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

Step 22.4

Combine and simplify the denominator.

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Step 22.4.1

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

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

Step 22.4.2

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

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

Step 22.4.3

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

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

Step 22.4.4

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

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

Step 22.4.5

Add $1$ and $1$ .

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

Step 22.4.6

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

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Step 22.4.6.1

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

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

Step 22.4.6.2

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

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

Step 22.4.6.3

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

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

Step 22.4.6.4

Cancel the common factor of $2$ .

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Step 22.4.6.4.1

Cancel the common factor.

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

Step 22.4.6.4.2

Rewrite the expression.

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

Step 22.4.6.5

Evaluate the exponent.

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

Step 22.5

Cancel the common factor of $2$ .

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Step 22.5.1

Cancel the common factor.

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

Step 22.5.2

Divide $\sqrt{2}$ by $1$ .

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

Step 22.6

Simplify the expression.

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Step 22.6.1

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

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

Step 22.6.2

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

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

Step 22.6.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

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

Step 22.7

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

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Step 22.7.1

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

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

Step 22.7.2

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

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

Step 22.7.3

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

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

Step 22.7.4

Cancel the common factor of $2$ .

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Step 22.7.4.1

Cancel the common factor.

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

Step 22.7.4.2

Rewrite the expression.

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

Step 22.7.5

Evaluate the exponent.

$- 2 \cdot 2 \tan (\frac{3 π}{4})$

Step 22.8

Multiply $- 2$ by $2$ .

$- 4 \tan (\frac{3 π}{4})$

Step 22.9

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

$- 4 (- \tan (\frac{π}{4}))$

Step 22.10

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$- 4 (- 1 \cdot 1)$

Step 22.11

Multiply $- 4 (- 1 \cdot 1)$ .

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Step 22.11.1

Multiply $- 1$ by $1$ .

$- 4 \cdot - 1$

Step 22.11.2

Multiply $- 4$ by $- 1$ .

$4$

Step 23

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

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

Step 24

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

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Step 24.1

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

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

Step 24.2

Simplify the result.

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Step 24.2.1

Simplify each term.

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Step 24.2.1.1

Cancel the common factor of $2$ .

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Step 24.2.1.1.1

Factor $2$ out of $4$ .

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

Step 24.2.1.1.2

Cancel the common factor.

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

Step 24.2.1.1.3

Rewrite the expression.

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

Step 24.2.1.2

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

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

Step 24.2.1.3

The exact value of $\tan (\frac{π}{4})$ is $1$ .

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

Step 24.2.1.4

Multiply $- (- 1 \cdot 1)$ .

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Step 24.2.1.4.1

Multiply $- 1$ by $1$ .

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

Step 24.2.1.4.2

Multiply $- 1$ by $- 1$ .

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

Step 24.2.2

The final answer is $\frac{3 π}{2} + 1$ .

$y = \frac{3 π}{2} + 1$

Step 25

Evaluate the second derivative at $x = \frac{5 π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2 \sec^{2} (\frac{5 π}{4}) \tan (\frac{5 π}{4})$

Step 26

Evaluate the second derivative.

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Step 26.1

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

$- 2 {(- \sec (\frac{π}{4}))}^{2} \tan (\frac{5 π}{4})$

Step 26.2

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$- 2 {(- \frac{2}{\sqrt{2}})}^{2} \tan (\frac{5 π}{4})$

Step 26.3

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

$- 2 {(- (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))}^{2} \tan (\frac{5 π}{4})$

Step 26.4

Combine and simplify the denominator.

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Step 26.4.1

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

$- 2 {(- \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.2

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

$- 2 {(- \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.3

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

$- 2 {(- \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.4

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

$- 2 {(- \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.5

Add $1$ and $1$ .

$- 2 {(- \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.6

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

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Step 26.4.6.1

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

$- 2 {(- \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.6.2

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

$- 2 {(- \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.6.3

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

$- 2 {(- \frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.6.4

Cancel the common factor of $2$ .

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Step 26.4.6.4.1

Cancel the common factor.

$- 2 {(- \frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.6.4.2

Rewrite the expression.

$- 2 {(- \frac{2 \sqrt{2}}{2^{1}})}^{2} \tan (\frac{5 π}{4})$

Step 26.4.6.5

Evaluate the exponent.

$- 2 {(- \frac{2 \sqrt{2}}{2})}^{2} \tan (\frac{5 π}{4})$

Step 26.5

Cancel the common factor of $2$ .

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Step 26.5.1

Cancel the common factor.

$- 2 {(- \frac{2 \sqrt{2}}{2})}^{2} \tan (\frac{5 π}{4})$

Step 26.5.2

Divide $\sqrt{2}$ by $1$ .

$- 2 {(- \sqrt{2})}^{2} \tan (\frac{5 π}{4})$

Step 26.6

Simplify the expression.

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Step 26.6.1

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

$- 2 ({(- 1)}^{2} {\sqrt{2}}^{2}) \tan (\frac{5 π}{4})$

Step 26.6.2

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

$- 2 (1 {\sqrt{2}}^{2}) \tan (\frac{5 π}{4})$

Step 26.6.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$- 2 {\sqrt{2}}^{2} \tan (\frac{5 π}{4})$

Step 26.7

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

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Step 26.7.1

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

$- 2 {(2^{\frac{1}{2}})}^{2} \tan (\frac{5 π}{4})$

Step 26.7.2

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

$- 2 \cdot 2^{\frac{1}{2} \cdot 2} \tan (\frac{5 π}{4})$

Step 26.7.3

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

$- 2 \cdot 2^{\frac{2}{2}} \tan (\frac{5 π}{4})$

Step 26.7.4

Cancel the common factor of $2$ .

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Step 26.7.4.1

Cancel the common factor.

$- 2 \cdot 2^{\frac{2}{2}} \tan (\frac{5 π}{4})$

Step 26.7.4.2

Rewrite the expression.

$- 2 \cdot 2^{1} \tan (\frac{5 π}{4})$

Step 26.7.5

Evaluate the exponent.

$- 2 \cdot 2 \tan (\frac{5 π}{4})$

Step 26.8

Multiply $- 2$ by $2$ .

$- 4 \tan (\frac{5 π}{4})$

Step 26.9

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

$- 4 \tan (\frac{π}{4})$

Step 26.10

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$- 4 \cdot 1$

Step 26.11

Multiply $- 4$ by $1$ .

$- 4$

Step 27

$x = \frac{5 π}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{5 π}{4}$ is a local maximum

Step 28

Find the y-value when $x = \frac{5 π}{4}$ .

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Step 28.1

Replace the variable $x$ with $\frac{5 π}{4}$ in the expression.

$f (\frac{5 π}{4}) = 2 (\frac{5 π}{4}) - \tan (\frac{5 π}{4})$

Step 28.2

Simplify the result.

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Step 28.2.1

Simplify each term.

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Step 28.2.1.1

Cancel the common factor of $2$ .

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Step 28.2.1.1.1

Factor $2$ out of $4$ .

$f (\frac{5 π}{4}) = 2 (\frac{5 π}{2 (2)}) - \tan (\frac{5 π}{4})$

Step 28.2.1.1.2

Cancel the common factor.

$f (\frac{5 π}{4}) = 2 (\frac{5 π}{2 \cdot 2}) - \tan (\frac{5 π}{4})$

Step 28.2.1.1.3

Rewrite the expression.

$f (\frac{5 π}{4}) = \frac{5 π}{2} - \tan (\frac{5 π}{4})$

Step 28.2.1.2

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

$f (\frac{5 π}{4}) = \frac{5 π}{2} - \tan (\frac{π}{4})$

Step 28.2.1.3

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$f (\frac{5 π}{4}) = \frac{5 π}{2} - 1 \cdot 1$

Step 28.2.1.4

Multiply $- 1$ by $1$ .

$f (\frac{5 π}{4}) = \frac{5 π}{2} - 1$

Step 28.2.2

The final answer is $\frac{5 π}{2} - 1$ .

$y = \frac{5 π}{2} - 1$

Step 29

These are the local extrema for $f (x) = 2 x - \tan (x)$ .

$(\frac{π}{4}, \frac{π}{2} - 1)$ is a local maxima

$(\frac{7 π}{4}, \frac{7 π}{2} + 1)$ is a local minima

$(\frac{3 π}{4}, \frac{3 π}{2} + 1)$ is a local minima

$(\frac{5 π}{4}, \frac{5 π}{2} - 1)$ is a local maxima

Step 30