Question

match the graph of the function with the graph of its derivative.

Explanation:

Step1: Analyze increasing - decreasing intervals

When the function is increasing, the derivative is positive. When the function is decreasing, the derivative is negative. The given function is increasing on $(-\infty,0)$ and decreasing on $(0,\infty)$. So the derivative is positive for $x < 0$ and negative for $x>0$.

Step2: Analyze slope at $x = 0$

The function has a horizontal tangent at $x = 0$. So the value of the derivative at $x=0$ is $0$.

Answer:

The derivative graph will be a curve that is above the $x$-axis for $x<0$, cross the $x$-axis at $x = 0$, and be below the $x$-axis for $x>0$.