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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

Use the first-derivative test to determine the local extrema of each function $f$ in Exercises $39 - 50$ . Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f (x) = a r c \tan x$ .

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f (x) = \frac{{(x - 1)}^{2}}{x + 2}$

Use the definitions of increasing and decreasing to argue that $f (x) = x^{4}$ is decreasing on $(- \infty, 0]$ and increasing on $[0, \infty)$ . Then use derivatives to argue the same thing.

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f (x) = x^{2} + 3 x$

For each set of sign charts in Exercises 53–62, sketch a possible graph of f.

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Theorem

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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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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Applied Mathematics

Geometry

Pure Maths

Discrete Mathematics

Logic and Functions

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