question. determine which cubic function (a, b, c, or d) would give the slopes of the tangent lines to the…

Explanation:

Step1: Recall derivative concept

The derivative of a function gives the slope of the tangent line at a point. The cubic - function that gives the slopes of the tangent lines to the quartic function is its derivative.

Step2: Analyze slope behavior

When the quartic function (dotted curve) has a horizontal tangent (slope = 0), the cubic - function (the candidate curves a, b, c, d) should cross the x - axis. Also, when the quartic function is increasing, the cubic function (its derivative) should be positive, and when the quartic function is decreasing, the cubic function should be negative.

Step3: Examine curves

By observing the behavior of the dotted quartic function: its increasing and decreasing intervals and where it has horizontal tangents, we can see that curve c (blue) is the derivative of the quartic function. When the quartic function is increasing (going up from left - to - right), the blue curve is above the x - axis (positive), and when the quartic function is decreasing, the blue curve is below the x - axis (negative). Also, at the points where the quartic function has horizontal tangents, the blue curve crosses the x - axis.

Answer:

c (blue)