Calculus Examples

$f (x) = 5 \cos (2 x) - 12 \sin (2 x) + 3$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $5 \cos (2 x) - 12 \sin (2 x) + 3$ with respect to $x$ is $\frac{d}{d x} [5 \cos (2 x)] + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [5 \cos (2 x)] + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.2

Evaluate $\frac{d}{d x} [5 \cos (2 x)]$ .

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Step 1.2.1

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

$5 \frac{d}{d x} [\cos (2 x)] + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 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 $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$5 (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]) + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.2.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$5 (- \sin (u_{1}) \frac{d}{d x} [2 x]) + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.2.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$5 (- \sin (2 x) \frac{d}{d x} [2 x]) + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.2.3

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

$5 (- \sin (2 x) (2 \frac{d}{d x} [x])) + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.2.4

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

$5 (- \sin (2 x) (2 \cdot 1)) + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.2.5

Multiply $2$ by $1$ .

$5 (- \sin (2 x) \cdot 2) + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.2.6

Multiply $2$ by $- 1$ .

$5 (- 2 \sin (2 x)) + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.2.7

Multiply $- 2$ by $5$ .

$- 10 \sin (2 x) + \frac{d}{d x} [- 12 \sin (2 x)] + \frac{d}{d x} [3]$

Step 1.3

Evaluate $\frac{d}{d x} [- 12 \sin (2 x)]$ .

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Step 1.3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 \sin (2 x)$ with respect to $x$ is $- 12 \frac{d}{d x} [\sin (2 x)]$ .

$- 10 \sin (2 x) - 12 \frac{d}{d x} [\sin (2 x)] + \frac{d}{d x} [3]$

Step 1.3.2

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

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Step 1.3.2.1

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

$- 10 \sin (2 x) - 12 (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [2 x]) + \frac{d}{d x} [3]$

Step 1.3.2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$- 10 \sin (2 x) - 12 (\cos (u_{2}) \frac{d}{d x} [2 x]) + \frac{d}{d x} [3]$

Step 1.3.2.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$- 10 \sin (2 x) - 12 (\cos (2 x) \frac{d}{d x} [2 x]) + \frac{d}{d x} [3]$

Step 1.3.3

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

$- 10 \sin (2 x) - 12 (\cos (2 x) (2 \frac{d}{d x} [x])) + \frac{d}{d x} [3]$

Step 1.3.4

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

$- 10 \sin (2 x) - 12 (\cos (2 x) (2 \cdot 1)) + \frac{d}{d x} [3]$

Step 1.3.5

Multiply $2$ by $1$ .

$- 10 \sin (2 x) - 12 (\cos (2 x) \cdot 2) + \frac{d}{d x} [3]$

Step 1.3.6

Move $2$ to the left of $\cos (2 x)$ .

$- 10 \sin (2 x) - 12 (2 \cdot \cos (2 x)) + \frac{d}{d x} [3]$

Step 1.3.7

Multiply $2$ by $- 12$ .

$- 10 \sin (2 x) - 24 \cos (2 x) + \frac{d}{d x} [3]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$- 10 \sin (2 x) - 24 \cos (2 x) + 0$

Step 1.4.2

Add $- 10 \sin (2 x) - 24 \cos (2 x)$ and $0$ .

$- 10 \sin (2 x) - 24 \cos (2 x)$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 10 \sin (2 x) - 24 \cos (2 x)$ with respect to $x$ is $\frac{d}{d x} [- 10 \sin (2 x)] + \frac{d}{d x} [- 24 \cos (2 x)]$ .

$f'' (x) = \frac{d}{d x} (- 10 \sin (2 x)) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.2

Evaluate $\frac{d}{d x} [- 10 \sin (2 x)]$ .

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Step 2.2.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 \sin (2 x)$ with respect to $x$ is $- 10 \frac{d}{d x} [\sin (2 x)]$ .

$f'' (x) = - 10 \frac{d}{d x} \sin (2 x) + \frac{d}{d x} (- 24 \cos (2 x))$

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 $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$f'' (x) = - 10 (\frac{d}{d u (1)} (\sin (u_{1})) \frac{d}{d x} (2 x)) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.2.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$f'' (x) = - 10 (\cos (u_{1}) \frac{d}{d x} (2 x)) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.2.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$f'' (x) = - 10 (\cos (2 x) \frac{d}{d x} (2 x)) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.2.3

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

$f'' (x) = - 10 (\cos (2 x) (2 \frac{d}{d x} (x))) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.2.4

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

$f'' (x) = - 10 (\cos (2 x) (2 \cdot 1)) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.2.5

Multiply $2$ by $1$ .

$f'' (x) = - 10 (\cos (2 x) \cdot 2) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.2.6

Move $2$ to the left of $\cos (2 x)$ .

$f'' (x) = - 10 (2 \cdot \cos (2 x)) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.2.7

Multiply $2$ by $- 10$ .

$f'' (x) = - 20 \cos (2 x) + \frac{d}{d x} (- 24 \cos (2 x))$

Step 2.3

Evaluate $\frac{d}{d x} [- 24 \cos (2 x)]$ .

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Step 2.3.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 \cos (2 x)$ with respect to $x$ is $- 24 \frac{d}{d x} [\cos (2 x)]$ .

$f'' (x) = - 20 \cos (2 x) - 24 \frac{d}{d x} \cos (2 x)$

Step 2.3.2

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

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Step 2.3.2.1

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

$f'' (x) = - 20 \cos (2 x) - 24 (\frac{d}{d u (2)} (\cos (u_{2})) \frac{d}{d x} (2 x))$

Step 2.3.2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$f'' (x) = - 20 \cos (2 x) - 24 (- \sin (u_{2}) \frac{d}{d x} (2 x))$

Step 2.3.2.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$f'' (x) = - 20 \cos (2 x) - 24 (- \sin (2 x) \frac{d}{d x} (2 x))$

Step 2.3.3

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

$f'' (x) = - 20 \cos (2 x) - 24 (- \sin (2 x) (2 \frac{d}{d x} (x)))$

Step 2.3.4

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

$f'' (x) = - 20 \cos (2 x) - 24 (- \sin (2 x) (2 \cdot 1))$

Step 2.3.5

Multiply $2$ by $1$ .

$f'' (x) = - 20 \cos (2 x) - 24 (- \sin (2 x) \cdot 2)$

Step 2.3.6

Multiply $2$ by $- 1$ .

$f'' (x) = - 20 \cos (2 x) - 24 (- 2 \sin (2 x))$

Step 2.3.7

Multiply $- 2$ by $- 24$ .

$f'' (x) = - 20 \cos (2 x) + 48 \sin (2 x)$

Step 3

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

$- 10 \sin (2 x) - 24 \cos (2 x) = 0$

Step 4

Divide each term in the equation by $\cos (2 x)$ .

$\frac{- 10 \sin (2 x)}{\cos (2 x)} + \frac{- 24 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 5

Separate fractions.

$\frac{- 10}{1} \cdot \frac{\sin (2 x)}{\cos (2 x)} + \frac{- 24 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 6

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$\frac{- 10}{1} \cdot \tan (2 x) + \frac{- 24 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 7

Divide $- 10$ by $1$ .

$- 10 \tan (2 x) + \frac{- 24 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 8

Cancel the common factor of $\cos (2 x)$ .

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Step 8.1

Cancel the common factor.

$- 10 \tan (2 x) + \frac{- 24 \cos (2 x)}{\cos (2 x)} = \frac{0}{\cos (2 x)}$

Step 8.2

Divide $- 24$ by $1$ .

$- 10 \tan (2 x) - 24 = \frac{0}{\cos (2 x)}$

Step 9

Separate fractions.

$- 10 \tan (2 x) - 24 = \frac{0}{1} \cdot \frac{1}{\cos (2 x)}$

Step 10

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$- 10 \tan (2 x) - 24 = \frac{0}{1} \cdot \sec (2 x)$

Step 11

Divide $0$ by $1$ .

$- 10 \tan (2 x) - 24 = 0 \sec (2 x)$

Step 12

Multiply $0$ by $\sec (2 x)$ .

$- 10 \tan (2 x) - 24 = 0$

Step 13

Add $24$ to both sides of the equation.

$- 10 \tan (2 x) = 24$

Step 14

Divide each term in $- 10 \tan (2 x) = 24$ by $- 10$ and simplify.

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Step 14.1

Divide each term in $- 10 \tan (2 x) = 24$ by $- 10$ .

$\frac{- 10 \tan (2 x)}{- 10} = \frac{24}{- 10}$

Step 14.2

Simplify the left side.

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Step 14.2.1

Cancel the common factor of $- 10$ .

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Step 14.2.1.1

Cancel the common factor.

$\frac{- 10 \tan (2 x)}{- 10} = \frac{24}{- 10}$

Step 14.2.1.2

Divide $\tan (2 x)$ by $1$ .

$\tan (2 x) = \frac{24}{- 10}$

Step 14.3

Simplify the right side.

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Step 14.3.1

Cancel the common factor of $24$ and $- 10$ .

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Step 14.3.1.1

Factor $2$ out of $24$ .

$\tan (2 x) = \frac{2 (12)}{- 10}$

Step 14.3.1.2

Cancel the common factors.

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Step 14.3.1.2.1

Factor $2$ out of $- 10$ .

$\tan (2 x) = \frac{2 \cdot 12}{2 \cdot - 5}$

Step 14.3.1.2.2

Cancel the common factor.

$\tan (2 x) = \frac{2 \cdot 12}{2 \cdot - 5}$

Step 14.3.1.2.3

Rewrite the expression.

$\tan (2 x) = \frac{12}{- 5}$

Step 14.3.2

Move the negative in front of the fraction.

$\tan (2 x) = - \frac{12}{5}$

Step 15

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

$2 x = \arctan (- \frac{12}{5})$

Step 16

Simplify the right side.

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Step 16.1

Evaluate $\arctan (- \frac{12}{5})$ .

$2 x = - 1.1760052$

Step 17

Divide each term in $2 x = - 1.1760052$ by $2$ and simplify.

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Step 17.1

Divide each term in $2 x = - 1.1760052$ by $2$ .

$\frac{2 x}{2} = \frac{- 1.1760052}{2}$

Step 17.2

Simplify the left side.

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Step 17.2.1

Cancel the common factor of $2$ .

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Step 17.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1.1760052}{2}$

Step 17.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1.1760052}{2}$

Step 17.3

Simplify the right side.

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Step 17.3.1

Divide $- 1.1760052$ by $2$ .

$x = - 0.5880026$

Step 18

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

$2 x = - 1.1760052 - (3.14159265)$

Step 19

Simplify the expression to find the second solution.

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Step 19.1

Add $2 π$ to $- 1.1760052 - (3.14159265)$ .

$2 x = - 1.1760052 - (3.14159265) + 2 π$

Step 19.2

The resulting angle of $1.96558744$ is positive and coterminal with $- 1.1760052 - (3.14159265)$ .

$2 x = 1.96558744$

Step 19.3

Divide each term in $2 x = 1.96558744$ by $2$ and simplify.

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Step 19.3.1

Divide each term in $2 x = 1.96558744$ by $2$ .

$\frac{2 x}{2} = \frac{1.96558744}{2}$

Step 19.3.2

Simplify the left side.

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Step 19.3.2.1

Cancel the common factor of $2$ .

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Step 19.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1.96558744}{2}$

Step 19.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.96558744}{2}$

Step 19.3.3

Simplify the right side.

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Step 19.3.3.1

Divide $1.96558744$ by $2$ .

$x = 0.98279372$

Step 20

The solution to the equation $2 x = - 1.1760052$ .

$x = - 0.5880026, 0.98279372$

Step 21

Evaluate the second derivative at $x = - 0.5880026$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 20 \cos (2 (- 0.5880026)) + 48 \sin (2 (- 0.5880026))$

Step 22

Simplify each term.

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Step 22.1

Multiply $2$ by $- 0.5880026$ .

$- 20 \cos (- 1.1760052) + 48 \sin (2 (- 0.5880026))$

Step 22.2

Multiply $2$ by $- 0.5880026$ .

$- 20 \cos (- 1.1760052) + 48 \sin (- 1.1760052)$

Step 23

$x = - 0.5880026$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 0.5880026$ is a local maximum

Step 24

Find the y-value when $x = - 0.5880026$ .

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Step 24.1

Replace the variable $x$ with $- 0.5880026$ in the expression.

$f (- 0.5880026) = 5 \cos (2 (- 0.5880026)) - 12 \sin (2 (- 0.5880026)) + 3$

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

Multiply $2$ by $- 0.5880026$ .

$f (- 0.5880026) = 5 \cos (- 1.1760052) - 12 \sin (2 (- 0.5880026)) + 3$

Step 24.2.1.2

Multiply $2$ by $- 0.5880026$ .

$f (- 0.5880026) = 5 \cos (- 1.1760052) - 12 \sin (- 1.1760052) + 3$

Step 24.2.2

The final answer is $5 \cos (- 1.1760052) - 12 \sin (- 1.1760052) + 3$ .

$y = 5 \cos (- 1.1760052) - 12 \sin (- 1.1760052) + 3$

Step 25

Evaluate the second derivative at $x = 0.98279372$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 20 \cos (2 (0.98279372)) + 48 \sin (2 (0.98279372))$

Step 26

Simplify each term.

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Step 26.1

Multiply $2$ by $0.98279372$ .

$- 20 \cos (1.96558744) + 48 \sin (2 (0.98279372))$

Step 26.2

Multiply $2$ by $0.98279372$ .

$- 20 \cos (1.96558744) + 48 \sin (1.96558744)$

Step 27

$x = 0.98279372$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0.98279372$ is a local minimum

Step 28

Find the y-value when $x = 0.98279372$ .

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Step 28.1

Replace the variable $x$ with $0.98279372$ in the expression.

$f (0.98279372) = 5 \cos (2 (0.98279372)) - 12 \sin (2 (0.98279372)) + 3$

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

Multiply $2$ by $0.98279372$ .

$f (0.98279372) = 5 \cos (1.96558744) - 12 \sin (2 (0.98279372)) + 3$

Step 28.2.1.2

Multiply $2$ by $0.98279372$ .

$f (0.98279372) = 5 \cos (1.96558744) - 12 \sin (1.96558744) + 3$

Step 28.2.2

The final answer is $5 \cos (1.96558744) - 12 \sin (1.96558744) + 3$ .

$y = 5 \cos (1.96558744) - 12 \sin (1.96558744) + 3$

Step 29

These are the local extrema for $f (x) = 5 \cos (2 x) - 12 \sin (2 x) + 3$ .

$(- 0.5880026, 5 \cos (- 1.1760052) - 12 \sin (- 1.1760052) + 3)$ is a local maxima

$(0.98279372, 5 \cos (1.96558744) - 12 \sin (1.96558744) + 3)$ is a local minima

Step 30