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Problem
Find all values of $a$ and $b$ that make the following function differentiable for all values of $x$: $$ f(x) = \begin{cases} \arctan(ax+b), x<0\\ \pi/4e^{\sin(bx)}, x \geq 0\\ \end{cases} $$

I thought I had this question figured out but it started to get more complicated than I think it should be. Any help would be greatly appreciated, Thank you.

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1 Answer 1

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It suffices to match the function at 0 and then evaluate the right and left hand limits for the derivatives and match these

$\arctan(b) = \frac{\pi}{4}$ implies that $b = 1$.

$\frac{a}{1+(ax+b)^2} = \frac{a}{1+1} $ set equal to $ \frac{\pi}{4}b =\frac{\pi}{4}$. So $a = \frac{\pi}{2}$

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Thanks so much, I was definitely over thinking that one –  Chance Oct 15 '12 at 5:21

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