Here is the problem I'm looking at:
Given $f: \mathbb{R} \to \mathbb{R}$ is differentiable, define the function
$ H(x) = \int_{-x}^x [f(t)+f(-t)] dt \text{ } \text{ } \text{ for all x}$
Find H''(x)
Now here's my crack at the solution. Is this right?
H'(x) = \displaystyle\frac{d}{dx} \displaystyle\int_{-x}^x f(t) + f(-t) dt = [x+(-x)] - [(-x) +x] = 0
H''(X) = \displaystyle\frac{d}{dx} H'(X) = \displaystyle\frac{d}{dx} 0 = 0