Question

determine all x - values for which the graph of f has a horizontal tangent line on the closed interval -9,9.

Explanation:

Step1: Recall tangent - slope relation

The slope of the tangent line to the graph of $y = f(x)$ is given by $f'(x)$. A horizontal tangent line has a slope of 0, so we need to find where $f'(x)=0$.

Step2: Identify points on graph

Visually inspect the graph of $y = f(x)$ on the interval $[-9,9]$. The graph has horizontal tangent lines at the points where the function has local maxima, local minima, or flat - regions.
From the graph, we can see that the $x$ - values where the slope of the tangent line is 0 (horizontal tangent) in the interval $[-9,9]$ are $x=-5,x = 2,x=6$.

Answer:

$x=-5,x = 2,x=6$