with the function f(x)= `(2x)/(1+5x^2)^2`  ,  -2`<= x<=` the absolute max value? when x equals? the absolute min value? when x equals? 



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Given `f(x)=(2x)/((1+5x^2)^2)` for `-2<=x<=2` find the absolute max and min:

Since the function is continuous and on a closed interval, we are guaranteed both a maximum and a minimum. We must check any critical values as well as the endpoints:

The critical values are where the first derivative is zero or fails to exist.

We use the quotient rule to find the first derivative:




The first derivative is continuous everywhere so we need only check its zeros:



The function's absolute maximum occurs at `x=sqrt(1/15)` while its absolute minimum occurs at `x=-sqrt(1/15)` regardless of the interval.


The graph:

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