# Find Where Increasing/Decreasing f(x)=3sin(x)+3cos(x) Find the derivative.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
The derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
The derivative of with respect to is .
Multiply by .
Set the derivative equal to .
Solve for .
Divide each term in the equation by .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Replace with an equivalent expression in the numerator.
Remove parentheses.
Apply the distributive property.
Multiply by .
Rewrite in terms of sines and cosines.
Apply the distributive property.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply .
Combine and .
Combine and .
Move the negative in front of the fraction.
Simplify each term.
Separate fractions.
Convert from to .
Divide by .
Multiply by .
Separate fractions.
Convert from to .
Divide by .
Multiply by .
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
The exact value of is .
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Simplify .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
The values which make the derivative equal to are .
Split into separate intervals around the values that make the derivative or undefined.
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify.
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify.
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Find Where Increasing/Decreasing f(x)=3sin(x)+3cos(x)

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