AP calculus, calculus

Question

Let f be the function given by f(x)=ln|x/1+x^2|. At what values of x does f have a relative maximum or a relative minimum? For each such x, use the first derivative test to determine whether f(x) is a relative maximum or a relative minimum.

Answer 1

To employ the 1st derivative test, we find the first derivative, set it = 0, solve to find all critical points, and lastly create a sign chart for f ‘ to determine what kind of critical points they are.

I’ll assume that the (1 + x²) is all in the denominator.

we need to use chain rule here:

f ‘ (x) = ((1+x²)-2x²)/(1+x²)²)((1+x²)/x)

= (1-x²)/(x(1+x²))

This function has zeros at x=±1 and is undefined at x=0.

f ‘ (x) goes from + to - at x=±1 and so f has local maxes at x=±1. f(0) is undefined, though f ‘ does change signs there.