How do you find the critical numbers for #f(x) = x2ln(x)# to determine the maximum and minimum?
The first derivative is given by
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To find the critical numbers for ( f(x) = x  2\ln(x) ), first, find the derivative of the function, then set it equal to zero and solve for ( x ).

Find the derivative of ( f(x) ) using the product rule: [ f'(x) = 1  \frac{2}{x} ]

Set ( f'(x) ) equal to zero and solve for ( x ): [ 1  \frac{2}{x} = 0 ]

Solve for ( x ): [ 1 = \frac{2}{x} ] [ x = 2 ]
So, the critical number is ( x = 2 ).
To determine whether it's a maximum or minimum, you can use the first or second derivative test.
Since ( f''(x) = \frac{2}{x^2} ), plug ( x = 2 ) into ( f''(x) ). [ f''(2) = \frac{2}{2^2} = \frac{1}{2} > 0 ]
Since ( f''(2) > 0 ), ( f(x) ) has a local minimum at ( x = 2 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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