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Calculus/Absolute extrema problem

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Question
Hi there,

I was wondering if you could guide me on how to do this question:

Find the absolute maximum value of the function f(x) = x + 1/x on the interval 1/2<=x<=4

I know that the derivative is f'(x) = 1 - 1/(x^2)

Thanks!
Nick

Answer
The absolute maximum occurs where the derivative is 0 or at one of the endpoints .
You found the derivative correctly. Set it equal to zero.
0 = 1-1/x^2
1/x^2 = 1
x^2 = 1
So x=1 is the only value that makes the derivative zero on the interval from 1/2 to 4

Evaluate the original function at x=1.

f (x) = x + 1/x

f (1) = 1+1/1 = 2

Now evaluate the function at the endpoints of the interval.

f (1/2) = 1/2 + 2 = 5/2

f (4) = 4 + 1/4 = 17/4

Finally , pick the largest value from

2 , 5/2 , 17/4

The absolute  maximum is 17/4

Calculus

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