# What are the local extrema, if any, of #f(x) =(lnx-1)^2 / x#?

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To find the local extrema of ( f(x) = \frac{{(\ln x - 1)^2}}{{x}} ), we need to take the derivative of ( f(x) ) and set it equal to zero.

First, let's find the derivative of ( f(x) ) using the quotient rule:

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

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

Now, we set ( f'(x) ) equal to zero and solve for ( x ):

[ 2(\ln x - 1) - (\ln x - 1)^2 = 0 ]

[ 2\ln x - 2 - (\ln^2 x - 2\ln x + 1) = 0 ]

[ 2\ln x - 2 - \ln^2 x + 2\ln x - 1 = 0 ]

[ \ln^2 x - 4\ln x + 3 = 0 ]

[ (\ln x - 1)(\ln x - 3) = 0 ]

[ \ln x = 1 \quad \text{or} \quad \ln x = 3 ]

[ x = e \quad \text{or} \quad x = e^3 ]

To determine whether these are local extrema, we can use the second derivative test.

[ f''(x) = \frac{{d}}{{dx}}\left(\frac{{2(\ln x - 1) - (\ln x - 1)^2}}{{x^2}}\right) ]

[ f''(x) = \frac{{2 - 2(\ln x - 1) \cdot \frac{1}{x} - 2(\ln x - 1) \cdot \frac{1}{x} + 2(\ln x - 1)^2 \cdot \frac{1}{x^2} - 2(\ln x - 1)^2 \cdot \frac{2}{x^3}}}{{x^2}} ]

[ f''(x) = \frac{{2 - 2\frac{1}{x} - 2\frac{1}{x} + 2(\ln x - 1) - 4(\ln x - 1) \cdot \frac{1}{x}}}{{x^2}} ]

[ f''(x) = \frac{{2 - 4\frac{1}{x} + 2(\ln x - 1) - 4(\ln x - 1) \cdot \frac{1}{x}}}{{x^2}} ]

[ f''(x) = \frac{{2 - 4\frac{1}{x} + 2\ln x - 2 - 4\ln x + 4}}{{x^2}} ]

[ f''(x) = \frac{{2\ln x - 2 - 4\frac{1}{x}}}{{x^2}} ]

At ( x = e ):

[ f''(e) = \frac{{2\ln e - 2 - 4\frac{1}{e}}}{{e^2}} = 0 ]

At ( x = e^3 ):

[ f''(e^3) = \frac{{2\ln(e^3) - 2 - 4\frac{1}{{e^3}}}}{{(e^3)^2}} = \frac{{6 - 2 - 4/e^3}}{{e^6}} ]

[ f''(e^3) = \frac{{4 - 4/e^3}}{{e^6}} > 0 ]

Therefore, ( x = e ) corresponds to a local minimum, and ( x = e^3 ) corresponds to a local maximum for the function ( f(x) ).

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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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