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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the local and absolute extreme values of the function on the given interval.

f ( x ) = x 1 x , [ 1 , 1 ]

To determine

To find: The local maximum and minimum and the absolute maximum and absolute minimum of the function f(x)=x1x on the interval [−1, 1].

Explanation

Given:

The function is, f(x)=x1x and the interval is, [−1, 1].

Calculation:

Obtain the first derivative of the given function.

f'(x)=(1)1xx21x=2(1x)x21x=23x21x

Set f(x)=0 and obtain the critical numbers.

23x21x=023x=0x=23

The critical number is x=23 which lies on the given interval [1,1].

Apply the extreme values of the given interval and the critical numbers in f(x).

Substitute x=1 in f(x),

f(1)=(1)1(1)=(1)2=2

Substitute x=23 in f(x),

f(23)=(23)1(23)=2313

Substitute x=1 in f(x),

f(1)=(1)1(1)=(1)0=0

Since the largest functional value is the absolute maximum and the smallest functional value is the absolute minimum, the absolute maximum of f(x) is 2313 and the absolute minimum of f(x) is 2.

Therefore, the absolute maxima of f(x) occurs at x=23 and the absolute minimum of f(x) is occurs at x=1

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