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

For

Determine the function's first derivative:

Thus, the function is stationary at this point and not increasing or decreasing.

If the second derivative is computed:

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To determine whether the function (f(x) = x(\ln(x))^2) is increasing or decreasing at (x = 1), we need to examine the sign of its derivative at (x = 1).

Using the product rule and chain rule, the derivative of (f(x)) is:

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

Now, to determine the behavior at (x = 1), we evaluate (f'(x)) at (x = 1):

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

Since (f'(1) = 0), we cannot conclude whether (f(x)) is increasing or decreasing at (x = 1) based solely on this information. Further analysis, such as examining the behavior of (f'(x)) in a neighborhood of (x = 1), would be necessary to determine the function's behavior at that point.

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To determine whether ( f(x) = x(\ln(x))^2 ) is increasing or decreasing at ( x = 1 ), we can examine the derivative of the function at that point.

First, find the derivative of ( f(x) ): [ f'(x) = (\ln(x))^2 + 2\ln(x) ]

Now, evaluate ( f'(1) ): [ f'(1) = (\ln(1))^2 + 2\ln(1) = 0 ]

Since ( f'(1) = 0 ), we cannot determine the monotonicity of ( f(x) ) at ( x = 1 ) using only the first derivative test. We need to further analyze the behavior of ( f(x) ) in the vicinity of ( x = 1 ).

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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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