`f(t) = root(3)(8 - t), [0, 8]` Find the absolute maximum and minimum values of f on the given interval

Textbook Question

Chapter 4, 4.1 - Problem 56 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

Posted on

`f(t)=root(3)(8-t)`

differentiating,

`f'(t)=(1/3)(8-t)^(1/3-1)(-1)`

`f'(t)=-1/(3(8-t)^(2/3))`

Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find the critical numbers that are in the interval and evaluate the function at the endpoints and at the critical numbers.

Now to find the critical numbers, solve for t for f'(t)=0.

`-1/(3(8-t)^(2/3))=0`

So, f'(t)=0 is not defined for any value of t.

Now evaluate the function at the end points of the interval.

`f(0)=root(3)(8-0)=2`

`f(8)=root(3)(8-8)=0`

So. the absolute maximum=2 at x=0

and absolute minimum=0 at x=8

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

Posted on

When t increases from 0 to 8, 8-t decreases from 8 to 0. And cubic root of 8-t also decreases monotonically from 2 to 0. Therefore the answer is: absolute minimum is 0 and absolute maximum is 2.

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