Question

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

Explanation:

Step1: Recall slope - tangent relationship

The slope of the tangent line to a function $y = h(x)$ at a point is given by the derivative $h'(x)$. If $h'(x)>0$, the function is increasing; if $h'(x)<0$, the function is decreasing; if $h'(x) = 0$, the function has a horizontal tangent.

Step2: Analyze increasing part for (a)

Looking at the graph, the function $h(x)$ is increasing (slope of tangent is positive) for $x>2$. We can choose $x = 3$.

Step3: Analyze decreasing part for (b)

The function $h(x)$ is decreasing (slope of tangent is negative) for $0

Step4: Analyze horizontal - tangent part for (c)

The function $h(x)$ has horizontal tangents (slope of tangent is zero) at the local maximum and local minimum points. From the graph, these occur at $x = 0$ and $x = 2$.

Answer:

(a) $x = 3$
(b) $x = 1$
(c) $x = 0,2$