# Is #f(x)=xlnx-x# concave or convex at #x=1#?

In this case, we have

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To determine whether ( f(x) = x \ln x - x ) is concave or convex at ( x = 1 ), we need to find the second derivative of ( f(x) ) and then evaluate it at ( x = 1 ).

The first derivative of ( f(x) ) is: [ f'(x) = \frac{d}{dx}(x \ln x - x) = \ln x + 1 - 1 = \ln x ]

The second derivative of ( f(x) ) is: [ f''(x) = \frac{d}{dx}(\ln x) = \frac{1}{x} ]

Evaluating ( f''(1) ): [ f''(1) = \frac{1}{1} = 1 ]

Since the second derivative ( f''(1) = 1 ) is positive, ( f(x) ) is convex at ( 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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