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Chapter 3: Q. 91 (page 277)

$P r o v e p a r t (b) o f T h e o r e m 3.10 : I f b o t h f a n d f' a r e d i f f e r e n t i a b l e o n a n i n t e r v a l I, a n d f'' i s n e g a t i v e o n I, t h e n f i s c o n c a v e d o w n o n I .$

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Step 1. Given Information.

Step 2. Theorem

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Most popular questions from this chapter

For each graph of f in Exercises 49–52, explain why f satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b] and approximate any values c ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.

Use a sign chart for $f^{'}$ to determine the intervals on which each function $f$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f (x) = \frac{e^{x}}{1 + e^{x}}$

Find the possibility graph of its derivative f'.

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f (x) = {(2 x - 1)}^{5}$

Use a sign chart for $f^{'}$ to determine the intervals on which each function $f$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$l n (x^{2} + 1)$

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Provepart(b)ofTheorem3.10:Ifbothfandf'aredifferentiableonanintervalI,andf''isnegativeonI,thenfisconcavedownonI.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Theorem

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$P r o v e p a r t (b) o f T h e o r e m 3.10 : I f b o t h f a n d f' a r e d i f f e r e n t i a b l e o n a n i n t e r v a l I, a n d f'' i s n e g a t i v e o n I, t h e n f i s c o n c a v e d o w n o n I .$