Question

below is the graph of a function f(x). (a) give the x - value of a point where the slope of the tangent line for f(x) is positive. x = (b) give the x - value of a point where the slope of the tangent line for f(x) is negative. x = (c) give the x - value for all of the points where the slope of the tangent line for f(x) is zero. x = (separate answers with a comma)

Explanation:

Step1: Recall slope - tangent line relationship

The slope of the tangent line to a function $y = f(x)$ at a point is related to the function's increasing or decreasing nature. A function $y=f(x)$ is increasing when the slope of its tangent line is positive, decreasing when the slope of its tangent line is negative, and has a horizontal tangent (slope = 0) at local extrema.

Step2: Find positive - slope point

Looking at the graph, the function $f(x)$ is increasing on the intervals $(-4,0)$ and $(4,5)$. We can choose $x=-2$ (any value in the increasing intervals will work).

Step3: Find negative - slope point

The function $f(x)$ is decreasing on the intervals $(-5,-4)$ and $(0,4)$. We can choose $x = 2$ (any value in the decreasing intervals will work).

Step4: Find zero - slope points

The function $f(x)$ has horizontal tangents (slope = 0) at the local maximum and local minimum points. From the graph, these occur at $x=-4,0,4$.

Answer:

(a) $x=-2$
(b) $x = 2$
(c) $x=-4,0,4$