Answered: dentify the function's local and…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 84E

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Question

Identify the function's local and absolute extreme values, if any, saying where they occur.

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h(x)=x-1/x^2+5x+10

Expert Solution

Step 1

$T h e g i v e n f u n c t i o n i s h (x) = \frac{x - 1}{x^{2} + 5 x + 10} F i r s t w e d r a w t h e g r a p h o f g i v e n f u n c t i o n h (x)$

Step 2

$H e r e h (x) = \frac{x - 1}{x^{2} + 5 x + 10} . . . (1) D i f f e r e n t i a t e e q u a t i o n (1) w . r . t . x, w e g e t \frac{d}{d x} h (x) = \frac{d}{d x} (\frac{x - 1}{x^{2} + 5 x + 10}) H e r e w e u s e Q u o t i e n t r u l e h' (x) = \frac{(x^{2} + 5 x + 10) \frac{d}{d x} (x - 1) - (x - 1) \frac{d}{d x} (x^{2} + 5 x + 10)}{{(x^{2} + 5 x + 10)}^{2}} h' (x) = \frac{(x^{2} + 5 x + 10) (1) - (x - 1) (2 x + 5)}{{(x^{2} + 5 x + 10)}^{2}} h' (x) = \frac{x^{2} + 5 x + 10 - 2 x^{2} - 3 x + 5}{{(x^{2} + 5 x + 10)}^{2}} h' (x) = \frac{- x^{2} + 2 x + 15}{{(x^{2} + 5 x + 10)}^{2}}$

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