Prove by induction that the function f(x) = exp(-1 / x^2) for x does not equal 0 and f(0) = 0 has derivatives of all orders at every point in R (reals) and that all of these derivatives vanish at x = 0. Hence this function is not given by its Taylor expansion about x = 0.

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Consider the function defined for every real number x edit The function is smooth The function f has ... View the full answer