# Use the First Derivative Test to find and classify each of the critical points of the function ...

## Question:

Use the First Derivative Test to find and classify each of the critical points of the function

{eq}f(x) = -\cos x - x \sin x {/eq} in the open interval (-3,3)

## First derivative test

The first derivative test examines a function's monotonic properties whether the function is increasing or decreasing focusing on a particular point or a particular range.

{eq}\frac{dy}{dx} = 0 {/eq} And where the first derivative is 0 there exists a critical point.

## Answer and Explanation:

{eq}f(x) = y = -\cos x - x \sin x \\ \frac{dy}{dx} = \sin x - x \cos x - \sin x = 0 \\ x \cos x = 0 \\ x = 0\ or\ \cos x = 0 \\ x = 0, -\frac{\pi}{2} , \frac{\pi}{2} {/eq} are three critical points between (-3,3).