(a) sketch a line that is tangent to the graph at (x = 3) on the graph above.\n(b) list all values of (x)…

Explanation:

Step1: Recall tangent - line concept for part (a)

To sketch the tangent line at (x = 3), draw a line that touches the curve at (x=3) and has the same slope as the curve at that point.

Step2: Identify slope - zero points for part (b)

The slope of the tangent line to (y = f(x)) is (0) at the local maxima and minima. From the graph, the values of (x) where the slope of the tangent line is (0) are the (x) - coordinates of the peak and the troughs. The local maximum occurs between (x = 2) and (x = 3) and the local minima occur at (x = 1) and (x = 4). So (x=1, x = 4) are the values where the slope of the tangent line is (0).

Step3: Find (f^{\prime}(0)) for part (c)

(f^{\prime}(0)) is the slope of the tangent line to the graph of (y = f(x)) at (x = 0). The graph is a straight - line segment for (x\in[- 2,1]). The equation of the line passing through two points ((-2,3)) and ((1,0)) has slope (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 3}{1+2}=\frac{-3}{3}=-1). Since (x = 0) lies on this line segment, (f^{\prime}(0)=-1).

Answer:

(a) Sketch a line touching the curve at (x = 3) with the same slope as the curve at that point. (b) (x = 1,x = 4) (c) (-1)