Question

(a) sketch a line that is tangent to the graph at (x = 3) on the graph above.
(b) list all values of (x) where the slope of the tangent line to the graph of (y = f(x)) would be 0.
(c) find (f(0)).

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) \(-1\)