# How do you find the critical numbers for #f(x) = x^(-2)ln(x)# to determine the maximum and minimum?

Plug in these numbers into the initial equation:

We'll need to use limits for this:

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To find the critical numbers of ( f(x) = x^{-2}\ln(x) ), you need to first find its derivative, then set it equal to zero and solve for ( x ). After finding the critical numbers, you can determine whether they correspond to maximum or minimum points by using the first or second derivative test.

The derivative of ( f(x) = x^{-2}\ln(x) ) can be found using the product rule and the chain rule. The derivative is ( f'(x) = -2x^{-3}\ln(x) + x^{-3} ).

Setting ( f'(x) ) equal to zero and solving for ( x ), we have ( -2x^{-3}\ln(x) + x^{-3} = 0 ). Multiplying through by ( x^3 ), we get ( -2\ln(x) + 1 = 0 ).

Solving ( -2\ln(x) + 1 = 0 ) for ( x ), we find ( x = e^{1/2} ) or approximately ( x = 1.64872 ).

To determine whether ( x = e^{1/2} ) corresponds to a maximum or minimum, you can use the second derivative test or analyze the behavior of the function around that point.

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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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