Question

determine the graph of f from the graph of f. choose the correct graph of f.

Explanation:

Step1: Analyze slope of f(x) (left segment)

The left segment of $f(x)$ goes from $(-8,8)$ to $(0,0)$. Calculate slope:
$$m_1 = \frac{0-8}{0-(-8)} = \frac{-8}{8} = -1$$
So $f'(x) = -1$ for $x \in (-8,0)$.

Step2: Analyze slope of f(x) (middle segment)

The middle segment of $f(x)$ goes from $(0,0)$ to $(2,2)$. Calculate slope:
$$m_2 = \frac{2-0}{2-0} = \frac{2}{2} = 1$$
So $f'(x) = 1$ for $x \in (0,2)$.

Step3: Analyze slope of f(x) (right segment)

The right segment of $f(x)$ goes from $(2,2)$ to $(8,-4)$. Calculate slope:
$$m_3 = \frac{-4-2}{8-2} = \frac{-6}{6} = -1$$
So $f'(x) = -1$ for $x \in (2,8)$.

Step4: Match to option

The graph of $f'(x)$ is a piecewise constant function: $-1$ on $(-8,0)$, $1$ on $(0,2)$, $-1$ on $(2,8)$, with open circles at the break points $x=0,2$. This matches option C.

Answer:

C. (The graph with $y=-1$ on $(-8,0)$ and $(2,8)$, $y=1$ on $(0,2)$, with open circles at the segment endpoints)