Find Where Increasing/Decreasing f(x)=3sin(x)+3cos(x)

$f (x) = 3 \sin (x) + 3 \cos (x)$

Find the derivative.

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

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

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

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

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

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

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

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

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

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

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

$3 \cos (x) + 3 (- \sin (x))$

Multiply $- 1$ by $3$ .

$3 \cos (x) - 3 \sin (x)$

Set the derivative equal to $0$ .

$3 \cos (x) - 3 \sin (x) = 0$

Solve for $x$ .

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

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

Factor $3$ out of $3 \cos (x) - 3 \sin (x)$ .

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Factor $3$ out of $3 \cos (x)$ .

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

Factor $3$ out of $- 3 \sin (x)$ .

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

Factor $3$ out of $3 (\cos (x)) + 3 (- \sin (x))$ .

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

Replace $\cos (x)$ with an equivalent expression in the numerator.

$3 (\cos (x) - \sin (x)) \cdot \sec (x) = \frac{0}{\cos (x)}$

Remove parentheses.

$3 (\cos (x) - \sin (x)) \cdot \sec (x) = \frac{0}{\cos (x)}$

Apply the distributive property.

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

Multiply $- 1$ by $3$ .

$(3 \cos (x) - 3 \sin (x)) \cdot \sec (x) = \frac{0}{\cos (x)}$

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Apply the distributive property.

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

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

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Factor $\cos (x)$ out of $3 \cos (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 3 \sin (x) \frac{1}{\cos (x)}$ .

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Combine $\frac{1}{\cos (x)}$ and $- 3$ .

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

Combine $\frac{- 3}{\cos (x)}$ and $\sin (x)$ .

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

Move the negative in front of the fraction.

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

Simplify each term.

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Separate fractions.

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

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

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

Divide $3$ by $1$ .

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

Multiply $3$ by $- 1$ .

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

Separate fractions.

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

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

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

Divide $0$ by $1$ .

$3 - 3 \tan (x) = 0 \sec (x)$

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

$3 - 3 \tan (x) = 0$

Subtract $3$ from both sides of the equation.

$- 3 \tan (x) = - 3$

Divide each term by $- 3$ and simplify.

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Divide each term in $- 3 \tan (x) = - 3$ by $- 3$ .

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

Cancel the common factor of $- 3$ .

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Cancel the common factor.

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

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

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

Divide $- 3$ by $- 3$ .

$\tan (x) = 1$

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

$x = \arctan (1)$

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

$x = \frac{π}{4}$

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.

$x = π + \frac{π}{4}$

Simplify $π + \frac{π}{4}$ .

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{π}{1} \cdot \frac{4}{4} + \frac{π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{π \cdot 4}{1 \cdot 4} + \frac{π}{4}$

Multiply $4$ by $1$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Simplify the numerator.

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Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Find the period.

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

Solve the equation.

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

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

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

The values which make the derivative equal to $0$ are $\frac{π}{4}, \frac{5 π}{4}$ .

$\frac{π}{4}, \frac{5 π}{4}$

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, \frac{π}{4}) \cup (\frac{π}{4}, \frac{5 π}{4}) \cup (\frac{5 π}{4}, \infty)$

Substitute a value from the interval $(- \infty, \frac{π}{4})$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $- 0.2146018$ in the expression.

$f' (- 0.2146018) = 3 \cos (- 0.2146018) - 3 \sin (- 0.2146018)$

The final answer is $3 \cos (- 0.2146018) - 3 \sin (- 0.2146018)$ .

$3 \cos (- 0.2146018) - 3 \sin (- 0.2146018)$

At $x = - 0.2146018$ the derivative is $3 \cos (- 0.2146018) - 3 \sin (- 0.2146018)$ . Since this is negative, the function is decreasing on $(- \infty, \frac{π}{4})$ .

Decreasing on $(- \infty, \frac{π}{4})$ since $f' (x) < 0$

Substitute a value from the interval $(0.7853982, \frac{5 π}{4})$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $2.3561946$ in the expression.

$f' (2.3561946) = 3 \cos (2.3561946) - 3 \sin (2.3561946)$

The final answer is $3 \cos (2.3561946) - 3 \sin (2.3561946)$ .

$3 \cos (2.3561946) - 3 \sin (2.3561946)$

Simplify.

$- 4.24264068$

At $x = 2.3561946$ the derivative is $- 4.24264068$ . Since this is negative, the function is decreasing on $(0.7853982, \frac{5 π}{4})$ .

Decreasing on $(\frac{π}{4}, \frac{5 π}{4})$ since $f' (x) < 0$

Substitute a value from the interval $(\frac{5 π}{4}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $4.926991$ in the expression.

$f' (4.926991) = 3 \cos (4.926991) - 3 \sin (4.926991)$

The final answer is $3 \cos (4.926991) - 3 \sin (4.926991)$ .

$3 \cos (4.926991) - 3 \sin (4.926991)$

Simplify.

$3.57005945$

At $x = 4.926991$ the derivative is $3.57005945$ . Since this is positive, the function is increasing on $(\frac{5 π}{4}, \infty)$ .

Increasing on $(\frac{5 π}{4}, \infty)$ since $f' (x) > 0$

List the intervals on which the function is increasing and decreasing.

Increasing on: $(\frac{5 π}{4}, \infty)$

Decreasing on: $(- \infty, \frac{π}{4}), (\frac{π}{4}, \frac{5 π}{4})$

Find Where Increasing/Decreasing f(x)=3sin(x)+3cos(x)

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