Question

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

Explanation:

Step1: Understand horizontal tangent

A horizontal tangent line occurs where the derivative \( f'(x) = 0 \), which corresponds to local maxima, local minima, or points of inflection with a horizontal slope on the graph of \( f(x) \). From the graph, we identify the \( x \)-values where the slope of the tangent is zero (the graph has a "peak" or "valley" or a flat point).

Step2: Identify critical points

Looking at the graph:

• There's a local maximum (peak) at \( x = -2 \) (since the graph changes from increasing to decreasing here).
• There's a local minimum (valley) at \( x = 2 \) (since the graph changes from decreasing to increasing here).
• There's another local maximum (peak) at \( x = 4 \) (since the graph changes from increasing to decreasing here). Also, we check the domain \([-9,9]\); these points lie within this interval.

Answer:

\( x = -2 \), \( x = 2 \), \( x = 4 \)