# What is the absolute extrema of the function: #2x/(x^2 +1)# on closed interval [-2,2]?

So, let's find the local extrema:

if

Now let's find the ordinate of the points at the extrema of the interval:

So the candidates are:

graph{2x/(x^2 +1) [-2, 2, -5, 5]}

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To find the absolute extrema of the function ( f(x) = \frac{2x}{x^2 + 1} ) on the closed interval ([-2, 2]), we follow these steps:

- Find the critical points of the function within the interval by finding where the derivative equals zero or is undefined.
- Evaluate the function at the critical points and the endpoints of the interval.
- The highest and lowest values obtained from these evaluations will be the absolute maximum and minimum, respectively.

Derivative of ( f(x) ): [ f'(x) = \frac{2(x^2 + 1) - 2x(2x)}{(x^2 + 1)^2} = \frac{2 - 2x^2 + 2x^2}{(x^2 + 1)^2} = \frac{2}{(x^2 + 1)^2} ]

Critical points: [ f'(x) = 0 ] [ \frac{2}{(x^2 + 1)^2} = 0 ] [ 2 = 0 ] [ \text{No critical points} ]

The derivative is never undefined, so there are no critical points within the interval.

Now, we evaluate the function at the endpoints and the critical points (which we found none): [ f(-2) = \frac{2(-2)}{(-2)^2 + 1} = \frac{-4}{5} ] [ f(2) = \frac{2(2)}{(2)^2 + 1} = \frac{4}{5} ]

Since there are no critical points within the interval, the only potential extrema occur at the endpoints.

Therefore, the absolute extrema of ( f(x) ) on the interval ([-2, 2]) are: Absolute maximum: ( f(2) = \frac{4}{5} ) Absolute minimum: ( f(-2) = \frac{-4}{5} )

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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