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

1 Answer
Jun 11, 2017

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

Explanation:

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.