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Q. 0

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Calculus

Found in: Page 247

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Read the section and make your own summary of the material.

Rolle's theorem and the mean value theorem.
Using critical points to calculate local extrema.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1. Given Information

Rolle's theorem and the mean value theorem.
Using critical points to calculate local extrema.

To study the mean value theorem of this section it is better to learn the definition of Local extrema, critical point, Rolle's theorem Therefore,

Definition: Local extrema

f has local maximum at x=c if $f (c) \geq f (x)$ for all nearby values of x=c.
f has local minimum at x=c if $f (c) \leq f (x)$ for all nearby values of x=c.

Rolle's theorem

if f is continuous on [a,b] and differentiable on (a,b) , and if $f (a) = f (b) = 0$ , then there exist at least one value $c \in (a, b)$ for which $f^{'} (c) = 0$ .

Mean value Theorem

If f is continuous on [a,b] and differentiable on (a,b) , and if $f (a) = f (b) = 0$ , then there exist at least one value $c \in (a, b)$ of such that

$f^{'} (c) = \frac{f (b) - f (a)}{b - a}$

local extrema can be found using the first derivative of the given function such that $f^{'} (x)_0$ , Every local extremum is a critical point , though every critical point is not a local extremum, In that case inflection points can be found.

Step 2. Conclusion

Given upper definitions are important for given Theorems.

Most popular questions for Math Textbooks

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.

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

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