Suppose that \[ f(x)=\left\{\begin{array}{ll} e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. \] a) Show that \( f \) is differentiable everywhere and find a formula for \( f^{\prime} \). b) Show that \( f^{\prime} \) is differentiable everywhere and find a formula for \( f^{\prime \prime} \). c) Suppose that \( k \) is a positive integer. Prove that \( \frac{d}{d x}\left(\frac{e^{-1 / x^{2}}}{x^{k}}\right) \) is a linear combination of functions of the form \( \frac{e^{-1 / x^{2}}}{x^{m}} \), where \( m \) is a positive integer. d) Hence deduce that \( f^{(n)}(0)=0 \) for every natural number \( n \). e) Write down the Maclaurin series for \( f \). Where does the Maclaurin series converge? Where does it converge to \( f ? \)
Added by Ahsan N.
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We can do this by considering the two cases in the definition of $f$. Case 1: $x \neq 0$. In this case, $f(x) = e^{-1/x^2}$. Using the chain rule, we have \[f'(x) = \frac{d}{dx} e^{-1/x^2} = e^{-1/x^2} \cdot \frac{d}{dx} (-1/x^2) = e^{-1/x^2} \cdot Show more…
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