Is #f(x)= cos(3x+(5pi)/4) # increasing or decreasing at #x=-pi/4 #?

Question

Is #f(x)= cos(3x+(5pi)/4)   # increasing or decreasing at #x=-pi/4   #?

Christopher Allers · Accepted Answer

decreasing

For using the firs derivative test, first get #f '(x)= -3sin(3x+(5pi)/4) # For #x = -pi/4, f '(-pi/4)=-3sin(pi/2)=-3#. Negative sign indicates that f(x) is decreasing at #x= -pi/4#

Ezekiel Alexander · Answer

undefined To determine if $ f(x) = \cos(3x + \frac{5\pi}{4}) $ is increasing or decreasing at $ x = -\frac{\pi}{4} $, we need to analyze the derivative of $ f(x) $ at that point.

Taking the derivative of $ f(x) $ with respect to $ x $, we get:

$$ f'(x) = -3\sin(3x + \frac{5\pi}{4}) $$

Evaluate $ f'(-\frac{\pi}{4}) $:

$$ f'(-\frac{\pi}{4}) = -3\sin(-\frac{3\pi}{4} + \frac{5\pi}{4}) = -3\sin\pi = 0 $$

Since $ f'(-\frac{\pi}{4}) = 0 $, we cannot determine whether $ f(x) $ is increasing or decreasing at $ x = -\frac{\pi}{4} $ using the first derivative test. Additional analysis, such as the second derivative test or analyzing the function around the point, may be required to determine the nature of the function at that point.