2. Intro to Derivatives

Suppose the graph of the derivative $f^{\prime }$ of a continuous function $f$ is shown and $f^{\prime }$ is continuous everywhere. Which of the following statements is necessarily true about the function $f$?

Suppose the graph of the derivative $f^{\prime }$ of a function $f$ is shown. At which of the following points does the function $f$ have a local maximum?

Given the graph of the function $f\left(x\right)$ below, which of the following graphs best represents its derivative $f^{\prime }\left(x\right)$?

Based on the graph $f\(\left\)(x\(\right\))$, describe all points where the derivative $f^{\(\prime\)}\(\left\)(x\(\right\))$would have a jump.

Based on the graph of $f\(\left\)(x\(\right\))$, describe the graph of the derivative $f^{\(\prime\)}\(\left\)(x\(\right\))$at the point $x=-1$.

Given that the graph of the derivative $f^{\prime }$ of a function $f$ is shown, which of the following statements is true about the function $f$?

Based on the graph of $f\(\left\)(x\(\right\))$, describe where the derivative curve $f^{\(\prime\)}\(\left\)(x\(\right\))$ would be below the x-axis.

Based on the graph of $f\(\left\)(x\(\right\))$, describe the graph of the derivative $f^{\(\prime\)}\(\left\)(x\(\right\))$ on the interval $\(\left\)(0,\(\infty\)\(\right\))$.