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Chapter 3: Q. 4 (page 313)

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

  • fis concave up or concave down on intervalI.

Short Answer

Expert verified

Ans: part (a). fis concave up on I if f'is increasing on I.

part (b). f is concave down on I if f' is decreasing on I.

Step by step solution

01

Step 1. Given Information:

fis concave up or concave down on interval I

02

Step 2. Graphical representation:

Consider the functions graphed next. On each graph four slopes are illustrated and

estimated. Notice that when f is concave up, its slopes increase from left to right, and when

f is concave down, its slopes decrease from left to right.

As we have already seen, a function increases where its derivative is positive.

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Determine the graph of a function f from the graph of its derivative f'.

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