match the graph of the function on the right with the graph of its derivative. choose the correct graph of…

Explanation:

Step1: Recall derivative - graph relationship

The derivative of a function gives the slope of the tangent line to the function at each point. When the function is increasing, the derivative is positive; when decreasing, negative; and at local extrema (peaks and valleys), the derivative is zero.

Step2: Analyze the given function graph

The given function has multiple local maxima and minima. At local maxima and minima, the derivative graph should cross the x - axis (y = 0). Also, when the function is increasing, the derivative is above the x - axis and when decreasing, below the x - axis.

Step3: Check each option

• Option A: The derivative graph does not correctly represent the increasing and decreasing intervals and extrema of the function graph.
• Option B: The derivative graph does not match the number of extrema and the behavior of the function graph in terms of increasing and decreasing intervals.
• Option C: The derivative graph does not match the shape and behavior of the function graph in terms of where the function is increasing and decreasing.
• Option D: The red - colored derivative graph correctly shows positive values when the blue - colored function graph is increasing, negative values when the function graph is decreasing, and crosses the x - axis at the local extrema of the function graph.

Answer:

D.