TITLE: Monotonicity of functions
QUESTION [8 upvotes]: Let $f(x) = xe^{x^2} + e^{-x^2}$
I'd like to prove that this function increases monotonically in the interval $(0,1)$.
I was able to do it by taking the derivative and proving it was greater than zero.
(though it took me quite some time to do that)
I wanted to know if there was any other more elegant method to do the same.

REPLY [3 votes]: The derivative is easy to compute, really, assuming I didn't screw up! Using an obvious abbreviation, it's
$$(1+2x^2)e^+-2xe^- > (1-2x+2x^2)e^+ =\left (2\left(x-\frac 1 2\right)^2 + \frac 1 2\right)e^+ > 0$$