# Find the absolute maximum and absolute minimum values of #f(x) = x^(1/3) e^(−x^2/8)# on the interval [−1, 4]?

The maximum is

Evaluating, we find that

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To find the absolute maximum and absolute minimum values of ( f(x) = x^{1/3} e^{-x^2/8} ) on the interval ([-1, 4]), we need to find critical points within the interval and evaluate the function at those points as well as at the endpoints.

First, find the derivative of ( f(x) ) using the product rule: [ f'(x) = \frac{1}{3}x^{-2/3} e^{-x^2/8} - \frac{x^{1/3}}{4} e^{-x^2/8} ]

Setting ( f'(x) = 0 ) to find critical points: [ \frac{1}{3}x^{-2/3} e^{-x^2/8} - \frac{x^{1/3}}{4} e^{-x^2/8} = 0 ]

Solving this equation will give us critical points. After solving, we find ( x = -1 ) and ( x = 4 ) as critical points within the interval ([-1, 4]).

Now, we evaluate ( f(x) ) at these critical points and at the endpoints ( x = -1 ) and ( x = 4 ), then compare the values to find the absolute maximum and minimum.

[ f(-1) = (-1)^{1/3} e^{-(-1)^2/8} ] [ f(4) = (4)^{1/3} e^{-4^2/8} ]

Calculate these values to determine the absolute maximum and minimum.

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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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