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Q. 39

Expert-verified

Calculus

Found in: Page 261

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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

Ans: The local extrema value of f(x) are $x = \{0, 4\}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

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

Step 2. Finding the derivative of the function

Rewriting the function by simplifying it,

f (x) = {(x - 2)}^{2} (x + 1) f (x) = (x^{2} - 4 x + 4) (1 + x) f (x) = x^{2} + x^{3} - 4 x - 4 x^{2} + 4 + 4 x f (x) = x^{3} - 3 x^{2} + 4 f' (x) = 3 x^{2} - 6 x l e t s, f' (x) = 0 ∴ 3 x^{2} - 6 x = 0 3 x (x - 2) = 0 x = {0, 2} s u b s t i t u t e x = - 1, 1, 3 i n t o f' (x) = 3 x^{2} - 6 x f' (- 1) = 3 {(- 1)}^{2} - 6 (- 1) = 9 > 0 f' (1) = 3 {(1)}^{2} - 6 (1) = - 3 < 0 f' (3) = 3 {(3)}^{2} - 6 (3) = 9 > 0

Step 3. Finding local extrema of the function on the number line(Sign chart):

$F r o m t h e g r a p h i d e n t i f y t h e l o c a l m i n i m u m a t x = 2 i n d i c a t e d o n t h e c h a r t w i t h t h e f i r s t d e r i v a t i v e t e s t, \sin c e f' (x) c h a n g e s s i g n f r o m n e g a t i v e t o p o s i t i v e a t x = 2 . f (0) = (0 - 2)^{2} (1 + 0) = 4 (l o c a l m a x i m u m) f (2) = (2 - 2)^{2} (1 + 2) = 0 (l o c a l m i n i m u m) ∴ t h e e x t r e m e v a l u e s o f t h e f u n c t i o n a r e x = {0, 4}$

Step 4. Verifying algebraic answers with graphs :

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