Calculus: Early Transcendentals 8th Edition

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

Chapter 4 - Section 4.1 - Maximum and Minimum Values - 4.1 Exercises: 1

Answer

Since, absolute minimum has smallest value of the function on the entire domain of the function, but local minimum has smallest value of the function when x is near c.

Work Step by Step

Step 1 of 1 The number $ f \left( c \right) $ is a local minimum value of $ f $ if $ f \left( c \right) \leq f \left( x \right) $ when x is near c .it is only an absolute minimum value of $ f $ if $ f \left( c \right) \leq f \left( x \right) $ for all x in the domain of $ f $. if $ f \left( c \right) $ is an absolute minimum value then it also must be a local minimum value. On the other hand, a number can be a local minimum value but not an absolute minimum value. Since, absolute minimum has smallest value of the function on the entire domain of the function, but local minimum has smallest value of the function when x is near c. So For Example ,$ f \left( 0 \right) =0 $ is a local minimum but not absolute minimum, and $ f \left( 3 \right) =-27 $ is both local minimum and absolute minimum.
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