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Find the Absolute Maximum and minimum values of f on the given interval. f(x) =...

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Find the Absolute Maximum and minimum values of f on the given interval.

f(x) = (x^2-4)/(x^2+4)  on  [-4,4]

 

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Find absolute maximum and minimum for `f(x)=(x^2-4)/(x^2+4)` on the interval [-4,4].

Since the function is continuous everywhere, there will be an absolute maximum and minimum on the closed interval.

The extrema can only occur at the endpoints of the interval or the critical points on the interval. To find the critical points we find where the first derivative is zero or fails to exist:

`f'(x)=((x^2+4)(2x)-(x^2-4)(2x))/((x^2+4)^2)` Using the quotient rule

`=(16x)/((x^2+4)^2)`

`f'(x)=0 ==>x=0`

So we check `f(-4),f(0),f(4):`

`f(-4)=f(4)=(16-4)/(16+4)=12/20=.6`

`f(0)=-4/4=-1`

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The minimum occurs at (0,-1) and the maximum at (-4,.6) and (4,.6)

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The graph:

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