Question

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

Explanation:

Step1: Analyze $f(x)$ (downward parabola)

$f(x)$ is a quadratic function: $f(x) = -ax^2 + c$ ($a>0$). Its derivative is $f'(x) = -2ax$, which is a linear function with negative slope, matching Graph B.

Step2: Analyze $g(x)$ (decreasing curve)

$g(x)$ is a decreasing, concave down function (e.g., $g(x) = -x^2$ for $x\geq0$, or a decaying curve). Its derivative $g'(x)$ is negative, decreasing, matching Graph D.

Step3: Analyze $h(x)$ (W-shaped curve)

$h(x)$ is an even function with two peaks. Its derivative $h'(x)$ is an odd function: negative left of 0, positive right of 0, crossing the x-axis at the minima of $h(x)$, matching Graph C.

Answer:

1. The graph of $f'(x)$ is: Graph B. A downward-sloping straight line through the origin
2. The graph of $g'(x)$ is: Graph D. A decreasing, concave-down curve below the x-axis
3. The graph of $h'(x)$ is: Graph C. An odd function crossing the x-axis twice, with a minimum left of 0 and maximum right of 0