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?

Analyzing Graphs

Each of the figures in Exercises 91 and 92 shows two graphs, the graph of a function 𝔂 = ƒ(x) together with the graph of its derivative ƒ'(x). Which graph is which? How do you know?

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Recovering a function from its derivative

b. Repeat part (a), assuming that the graph starts at (−2, 0) instead of (−2, 3).

Slopes on the graph of the tangent function Graph y = tan x and its derivative together on (−π/2, π/2). Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.

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 $f$?

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 $f$?

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

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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\))$.