Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 4 - Section 4.3 - How Derivatives Affect the Shape of a Graph - 4.3 Exercises: 67


$f'\gt0$ and $f''\lt0$.

Work Step by Step

Take $f(x)$ to represent the national deficit mentioned. The national deficit is increasing. So first, the graph of $f$ representing the national deficit must be increasing, which means $f'\gt0$. Now we notice the phrase 'is increasing at a decreasing rate'. That means it increases less rapidly than it did befor. The graph therefore goes from increasing steeply to increasing less steeply. This means all the tangent lines are above the graph $f$. Therefore $f$ is a concave downward. (The image below represents a model of the graph) Therefore, $f''\lt0$.
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