009A Sample Final 1, Problem 10
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Consider the following continuous function:
defined on the closed, bounded interval .
a) Find all the critical points for .
b) Determine the absolute maximum and absolute minimum values for on the interval .
Foundations: |
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Solution:
(a)
Step 1: |
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To find the critical point, first we need to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)} . |
Using the Product Rule, we have |
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Step 2: |
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Notice is undefined when . |
Now, we need to set . |
So, we get . |
We cross multiply to get . |
Solving, we get . |
Thus, the critical points for are and . |
(b)
Step 1: |
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We need to compare the values of at the critical points and at the endpoints of the interval. |
Using the equation given, we have and . |
Step 2: |
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Comparing the values in Step 1 with the critical points in (a), the absolute maximum value for is 32 |
and the absolute minimum value for is . |
Final Answer: |
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(a) and |
(b) The absolute minimum value for is |