Question

Proving that ln(x^2+1)-e^-x+1 is monotonically increasing on R

Answer 1

Note that 0≤(|x|-1)². Expanded out, this gives the so-called AM-GM inequality: 2|x|≤x²+1. It follows that 2x/(x²+1) ≥ -1. Also, for negative x, we have e^{-x} ≥1, so f'(x)≥0, as required. In fact, the inequality is easily seen to be strict.