Question

the graphs a (blue), b (red), and c (green) are the graphs of a function f(x) and its derivatives, f(x) and f(x). match the graph to the corresponding function.

1. a (blue)
2. b (red)
3. c (green)

Explanation:

Step1: Recall derivative - graph relationships

The derivative $f'(x)$ is zero at the local - extrema of $f(x)$. The second - derivative $f''(x)$ is zero at the inflection points of $f(x)$. Also, when $f'(x)>0$, $f(x)$ is increasing; when $f'(x)<0$, $f(x)$ is decreasing. When $f''(x)>0$, $f(x)$ is concave up; when $f''(x)<0$, $f(x)$ is concave down.

Step2: Analyze zero - crossing and slope

Look for the graph that has zero - crossings at the local extrema of another graph. Suppose graph $A$ has zero - crossings at the local extrema of graph $C$. Then $A$ could be $f'(x)$ and $C$ could be $f(x)$. Also, look at the concavity of a graph. If graph $B$ has zero - crossings at the inflection points of graph $C$, then $B$ could be $f''(x)$.

Step3: Match the graphs

Based on the above - mentioned derivative and concavity relationships: If graph $C$ has local extrema where graph $A$ crosses the $x$ - axis (i.e., $A = f'(x)$ when $C=f(x)$), and graph $B$ has zero - crossings where graph $C$ has inflection points (i.e., $B = f''(x)$ when $C=f(x)$).

Answer:

$f'(x)$: A (blue)
$f''(x)$: B (red)
$f(x)$: C (green)