Is f(x)=1/(x-1)-2xlnx concave or convex at x=0?
1 Answer
Jun 11, 2017
Neither. The function is undefined at
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.