# Is f(x)=cos(-x) increasing or decreasing at x=0?

Apr 6, 2018

$f \left(x\right) = \cos \left(- x\right)$ is neither increasing nor decreasing at $x = 0$

#### Explanation:

The first derivative provides the slope at each point of our function. A positive slope means the function is increasing and that a negative slope means that the function is decreasing.

Knowing this, we can find out if $\cos \left(- x\right)$ is increasing or decreasing at $\textcolor{red}{x = 0}$ by evaluating its first derivative at that point.

Before starting, we may want to simplify our function. Because $\cos \left(x\right)$ is an even function, we know that $\cos \left(- x\right) = \cos \left(x\right)$.

$\frac{d}{\mathrm{dx}} \left[\cos \left(x\right)\right]$

$= - \sin \left(x\right)$

Evaluate at $\textcolor{red}{x = 0}$:

$- \sin \left(\textcolor{red}{0}\right)$

$= - 0$

$= 0$

In this case, the slope is neither negative or positive, but 0. Hence, $f \left(x\right) = \cos \left(- x\right)$ is neither increasing nor decreasing at $\textcolor{red}{x = 0}$.