True/False: If $f(x)g(x) = x$, then it is possible that $f$ and $g$ are differentiable and $f(0)=g(0)=0$. If true, explain. If false, then give a counterexample.
Not even sure where to start, please help!
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4Do you know the product rule? – Daniel Fischer Oct 13 '13 at 01:56
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4The question gives you a big hint. If $f$ and $g$ had derivatives what could you do to that equation? – Spencer Oct 13 '13 at 01:56
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No, it is not possible:
By the product rule, we'd have
$$1 = (fg)'(0) = f(0) g'(0) + g(0) f'(0) = 0$$
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5Intuitively this is the observation that $f(x)g(x)$ vanishes to order at least two at $0$, whereas $h(x) = x$ only vanishes to order one at $0$. – Adrián Barquero Oct 13 '13 at 02:07