# What are the absolute extrema of #f(x)=(9x^(1/3))/(3x^2-1) in[2,9]#?

The absolute minimum is

The absolute maximum is

The absolute extrema of a function are the largest and smallest y-values of the function on a given domain. This domain may be given to us (as in this problem) or it might be the domain of the function itself. Even when we are given the domain, we must consider the domain of the function itself, in case it excludes any values of the domain we are given.

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To find the absolute extrema of ( f(x) = \frac{9x^{1/3}}{3x^2 - 1} ) on the interval ([2, 9]), we first need to find the critical points of the function within this interval. Then we evaluate the function at these critical points as well as at the endpoints of the interval, and the largest and smallest of these values will be the absolute maximum and minimum, respectively.

The critical points are found by setting the derivative of ( f(x) ) equal to zero and solving for ( x ). After finding the critical points, we evaluate the function at these points as well as at ( x = 2 ) and ( x = 9 ). The largest and smallest values among these will be the absolute maximum and minimum, respectively.

The derivative of ( f(x) ) is found using the quotient rule:

[ f'(x) = \frac{(3x^2 - 1) \cdot \frac{d}{dx}(9x^{1/3}) - 9x^{1/3} \cdot \frac{d}{dx}(3x^2 - 1)}{(3x^2 - 1)^2} ]

[ = \frac{(3x^2 - 1) \cdot \frac{9}{3}x^{-2/3} - 9x^{1/3} \cdot 6x}{(3x^2 - 1)^2} ]

[ = \frac{(3x^2 - 1) \cdot 3x^{-2/3} - 54x^{4/3}}{(3x^2 - 1)^2} ]

Setting ( f'(x) = 0 ), we solve for ( x ) to find the critical points. After finding the critical points, we evaluate ( f(x) ) at these points as well as at ( x = 2 ) and ( x = 9 ). The largest and smallest values among these will be the absolute maximum and minimum, respectively.

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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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