Concept explainers
Show that
To show:
Answer to Problem 1P
Solution:
Explanation of Solution
1) Concept:
Use Closed Interval Method to find the absolute maximum and absolute minimum values for the given function.
2) The Closed Interval Method:
To find the absolute maximum and minimum values of a continuous function
i) Find the values of
ii) Find the values of
iii) The largest of the values from the above steps is the absolute maximum value and the smallest of these values is the absolute minimum value.
3) Formula:
4) Calculation:
To show that
Using formula, above equation becomes
Now, let
The function
Take derivative of
Simplify.
To find critical numbers, equate
Subtract
Simplify.
Divide by
This is only true when,
Therefore, critical numbers are
To find the absolute maximum and minimum values
substitute critical numbers and endpoints in the function
Substitute.
Simplify.
Substitute.
Simplify.
Substitute,
Simplify.
Substitute.
Simplify.
By Closed Interval Method,
Therefore,
Therefore,
Conclusion:
Therefore,
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Calculus (MindTap Course List)
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- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage