# Is #f(x)=xln(x)^2# increasing or decreasing at #x=1#?

Determine every derivative:

The following calls for the chain rule:

Simplify.

graph{x (ln(x))^2 [-1.46, 5.565, 10.887, -3.16]}

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To determine if ( f(x) = x \ln(x)^2 ) is increasing or decreasing at ( x = 1 ), we need to evaluate the derivative of ( f(x) ) at ( x = 1 ) and check its sign.

Taking the derivative of ( f(x) ) with respect to ( x ) using the product rule, we get:

[ f'(x) = \ln(x)^2 + 2x \ln(x) \cdot \frac{1}{x} ]

Simplify the expression:

[ f'(x) = \ln(x)^2 + 2 \ln(x) ]

Evaluate ( f'(1) ):

[ f'(1) = \ln(1)^2 + 2 \ln(1) ]

Since ( \ln(1) = 0 ), we have:

[ f'(1) = 0^2 + 2 \cdot 0 = 0 ]

Since ( f'(1) = 0 ), we can't determine if ( f(x) ) is increasing or decreasing at ( x = 1 ) using the first derivative test. Further analysis, such as examining the behavior of the function on both sides of ( x = 1 ), would be needed to determine the behavior of the function at 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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