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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

15-17 Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer?

f ( x ) = 1 + 3 x 2 2 x 3

To determine

To find:

The local maximum and minimum values of f using both first and second derivative tests

Explanation

1) Concept:

By using first and second derivative tests

2) Tests:

a) First derivative test:

Suppose that c is a critical number of a continuous function f

i. If f' changes from positive to negative at c, then f has a local maximum at c

ii. If f' changes from negative to positive at c, then f has a local minimum at c

iii. If f' positive to the left and right of c or negative to the left and right of c then f has no local maximum or minimum at c

b) Second derivative test:

Suppose f" is continuous near c

i. If f'c=0 and f"(c)>o, then f has local minimum at c

ii. If f'c=0 and f"(c)<o, then f has local maximum at c

3) Formula:

i. Sum and difference rule:

ddxfx±gx=ddxfx±ddx(gx)

ii. Constant function rule:

ddxC=0

iii. Constant multiple rule:

ddxCfx=Cddx(fx)

iv. Power rule:

ddxxn=nxn-1

4) Given:

fx=1+3x2-2x3

5) Calculation:

Differentiate f with respect to x,

f'x=ddx(1+3x2-2x3)

By using sum and difference rule,

f'x=ddx(1)+ddx(3x2)-ddx(2x3)

By using constant function and constant multiple rule,

f'x=0+3

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