Is #f(x)=1/(x-1)-2xlnx# concave or convex at #x=0#?

Answer 1

Neither. The function is undefined at #x=0# since #ln0# is undefined.

However, you could take the limit as x approaches 0 of the concavity of the function. First, let's take the second derivative of the function.

#d/dx f(x) = -1/(x-1)^2-2x(1/x)-2lnx#
#=-1/(x-1)^2-2-2lnx#
#therefore (d^2y)/(dx^2) = 2/(x-1)^3 - 2/x#

Now we need to take the limit of this second derivative as x approaches 0. Since ln(x) is not defined for negatives, we only need to worry about when x approaches 0 from the positive direction.

#lim_(x->0)(2/(x-1)^3 - 2/x)#
#= 2/(0-1)^3 - lim_(x->0)2/x#
#= -2 - oo#
#= -oo#

So we can say that the function is concave down as x approaches 0.

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Answer 2

To determine the concavity or convexity of ( f(x) = \frac{1}{x - 1} - 2x \ln x ) at ( x = 0 ), we need to find the second derivative of the function and evaluate it at ( x = 0 ).

First, find the first derivative ( f'(x) ) using the quotient rule:

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

[ = \frac{-(x - 1)^{-2}}{1} - 2\left(\ln x + x \cdot \frac{1}{x}\right) ]

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

Now, find the second derivative ( f''(x) ) by differentiating ( f'(x) ):

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

[ = 2\frac{1}{(x - 1)^3} - \frac{2}{x} ]

Evaluate ( f''(0) ):

[ f''(0) = 2\frac{1}{(-1)^3} - \frac{2}{0} ]

[ = 2 - \text{undefined} ]

Since the second derivative is undefined at ( x = 0 ), the function ( f(x) = \frac{1}{x - 1} - 2x \ln x ) is neither concave nor convex at ( x = 0 ).

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Answer from HIX Tutor

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.

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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