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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

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    How do you justify the result for $H'(x)$?2011-11-30

2 Answers 2

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By the Second Fundamental Theorem of Calculus, $\frac{d}{dx}\int_a^x g(t)\,dt = g(x).$ Using the Chain Rule, \frac{d}{dx}\int_a^{h(x)}g(t)\,dt = g(h(x))h'(x).

So \begin{align*} H'(x) &= \frac{d}{dx}\int_{-x}^x (f(t)+f(-t))\,dt\\ &= \frac{d}{dx}\left(\int_0^x(f(t)+f(-t))\,dt - \int_0^{-x}(f(t)+f(-t))\,dt\right)\\ &= \left(f(x) + f(-x)\right) -\Bigl(\bigl( f(-x) + f(-(-x))\bigr)(-x)'\Bigr)\\ &= f(x) + f(-x) + f(-x) + f(x) \\ &= 2f(x) + 2f(-x). \end{align*} Therefore, H''(x) = \frac{d}{dx}(2f(x)+2f(-x)) = 2f'(x) + 2f'(-x)(-x)' = 2(f'(x) - f'(-x)).

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    Tha$n$k you very much, that makes sense now. I totally forgot that you had to use the chain rule.2011-11-30
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It is definitively not right. When you compute H'(x), you get \begin{align} H'(x) & = \lim_{h \to 0} \frac{H(x+h) - H(x)}{h} \\ & = \lim_{h \to 0} \frac{ \int_{-x-h}^{x+h} [f(t)+f(-t)] \, dx - \int_{-x}^x [f(t)+f(-t)] \, dx }{h} \\ & = \lim_{h \to 0} \frac{ \int_{x}^{x+h} [f(t)+f(-t)] \, dx + \int_{-x-h}^{-x} [f(t)+f(-t)] \, dx }h \\ & = [f(x) + f(-x)] + [f(-x) + f(-(-x))] = 2f(x) + 2f(-x). \\ \end{align} by the fundamental theorem of calculus. Thus H''(x) = 2f'(x) -2f'(-x). Hope that helps,

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    Thank you very much, that makes sense now; I knew mine didn't seem right.2011-11-30