the second derivative of the twice - differentiable function f is shown below on the domain (-9,9). the…
the second derivative of the twice - differentiable function f is shown below on the domain (-9,9). the graph of f has points of inflection at x = -5, x = -2, x = 3, indicated by small green circles. what inferences can be made about the graphs of f, f, and f on the interval (-2,0)? choose the best answer for each dropdown.
Answer
Explanation:
Step1: Analyze $f''$ on $(-2,0)$
On $(-2,0)$, the graph of $f''$ is negative and increasing. Since $f''<0$, the graph of $f'$ is decreasing. And since $f''$ is increasing, the graph of $f'$ is becoming less steep (concave - up). Also, since $f''<0$, the graph of $f$ is concave - down.
Step2: Summarize for $f$
The graph of $f$ is concave - down on $(-2,0)$ because $f''<0$.
Step3: Summarize for $f'$
The graph of $f'$ is decreasing and concave - up on $(-2,0)$ because $f''<0$ and $f''$ is increasing.
Step4: Summarize for $f''$
The graph of $f''$ is negative and increasing on $(-2,0)$ as seen from the given graph.
Answer:
On the interval $(-2,0)$: The graph of $f$ is concave - down, the graph of $f'$ is decreasing and concave - up, and the graph of $f''$ is negative and increasing.