the figure shows graphs of f, f, f, and f. identify each curve, and explain your choices.

Explanation:

Step1: Analyze number of critical points

The function $f$ has critical - points where $f' = 0$. Curve $b$ has two points where it has a horizontal tangent. So assume $b$ is $f$.

Step2: Check sign changes of derivative

The derivative $f'$ is positive when $f$ is increasing and negative when $f$ is decreasing. Curve $c$ is positive when $b$ is increasing and negative when $b$ is decreasing. So $c$ is $f'$.

Step3: Analyze concavity and second - derivative

The second - derivative $f''$ is positive when $f$ is concave up and negative when $f$ is concave down. Curve $d$ is positive when $b$ is concave up and negative when $b$ is concave down. So $d$ is $f''$.

Step4: Analyze the sign of the third - derivative

The third - derivative $f'''$ is related to the rate of change of the concavity of $f$. Curve $a$ is consistent with the rate of change of the concavity of $b$ (as determined by $d$). So $a$ is $f'''$.

Answer:

$f: b$, $f': c$, $f'': d$, $f''': a$