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

Jun 2, 2017

Increasing.

#### Explanation:

The derivative can tell you where an original function is increasing or decreasing. If the derivative is positive at your point of interest, then the original function is increasing at that point. If the derivative is negative at your point of interest, then the original function is decreasing at that point.

Step 1. Determine the derivative of $f \left(x\right)$

Requires the product rule

$\frac{d}{\mathrm{dx}} f \left(x\right) = 4 x \frac{d}{\mathrm{dx}} \left(\cos \left(3 x - \frac{5 \pi}{4}\right)\right) + \cos \left(3 x - \frac{5 \pi}{4}\right) \frac{d}{\mathrm{dx}} \left(4 x\right)$

$= - 12 x \sin \left(3 x - \frac{5 \pi}{4}\right) + 4 \cos \left(3 x - \frac{5 \pi}{4}\right)$

Step 2. Determine if $\frac{d}{\mathrm{dx}} f \left(x\right)$ is positive or negative at the desired point, $x = - \pi / 4$

At the value $x = - \frac{\pi}{4}$, the derivative is positive, which means that the original function, $f \left(x\right)$ is increasing at that point.