# Is the function concave up or down if #f(x)= (lnx)^2#?

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To determine if the function ( f(x) = (\ln x)^2 ) is concave up or down, we need to analyze its second derivative.

The first derivative of ( f(x) ) can be found as follows:

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

Now, let's find the second derivative:

[ f''(x) = \frac{d}{dx} \left(\frac{2\ln(x)}{x}\right) ]

Using the quotient rule, we get:

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

Simplify this expression:

[ f''(x) = \frac{2 - 2\ln(x)}{x^3} ]

To determine concavity, we need to examine the sign of ( f''(x) ). Since ( x ) is always positive in the domain of ( f(x) ), we only need to consider the sign of ( 2 - 2\ln(x) ).

( 2 - 2\ln(x) ) is negative for ( x > e ), where ( e ) is the Euler's number. This means ( f''(x) ) is negative for ( x > e ), indicating that the function ( f(x) ) is concave down in this interval.

Therefore, the function ( f(x) = (\ln x)^2 ) is concave down for ( x > e ).

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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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