How do you find the critical points for #f(x) = x  3ln(x)# and the local max and min?
graph{x3lnx [10, 10, 5, 5]}
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To find the critical points of ( f(x) = x  3 \ln(x) ), you first need to find its derivative and then solve for ( x ) where the derivative equals zero or is undefined.
First, compute the derivative of ( f(x) ): [ f'(x) = 1  \frac{3}{x} ]
Set ( f'(x) ) equal to zero and solve for ( x ): [ 1  \frac{3}{x} = 0 ] [ \frac{3}{x} = 1 ] [ x = 3 ]
Now, check for critical points where the derivative is undefined. Since the derivative involves a denominator, it's undefined when ( x = 0 ).
So, the critical points are ( x = 0 ) and ( x = 3 ).
To determine if these critical points are local minima, maxima, or neither, you can use the first derivative test or the second derivative test.

First Derivative Test:
 Determine the sign of the derivative around each critical point.
 If the derivative changes from positive to negative at a critical point, it's a local maximum.
 If the derivative changes from negative to positive at a critical point, it's a local minimum.

Second Derivative Test:
 Compute the second derivative of the function.
 Evaluate the second derivative at each critical point.
 If the second derivative is positive at a critical point, it's a local minimum.
 If the second derivative is negative at a critical point, it's a local maximum.
 If the second derivative is zero, the test is inconclusive.
After determining the nature of the critical points using one of these tests, you can identify the local maxima and minima accordingly.
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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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