Question

choose the best answer to complete the statements.
the graph of $f(x)$ is (choose one) $\boldsymbol{downarrow}$.
the graph of $g(x)$ is (choose one) $\boldsymbol{downarrow}$.
the graph of $h(x)$ is graph a $\boldsymbol{downarrow}$.

Explanation:

Step1: Analyze $f(x)$ derivative

The graph of $f(x)$ is a W-shaped even function: it decreases to the left root, increases to the peak, decreases to the right root, then increases. Its derivative $f'(x)$ will be negative, then positive, then negative, then positive, matching Graph D.

Step2: Analyze $g(x)$ derivative

The graph of $g(x)$ is a right-opening upward parabola: it decreases to the vertex, then increases. Its derivative $g'(x)$ is a linear function with negative slope then positive slope, crossing the x-axis, matching Graph B.

Step3: Confirm $h(x)$ derivative

The graph of $h(x)$ has horizontal segments (derivative 0), positive slope segment (derivative positive), negative slope segment (derivative negative), matching Graph A.

Answer:

1. The graph of $f'(x)$ is Graph D
2. The graph of $g'(x)$ is Graph B
3. The graph of $h'(x)$ is Graph A (correct as selected)