Chapter 4.4, Problem 91E

Let

$f(x)=\{\begin{array}{ll}{e}^{-1/x^2}\hfill & \text{if}x\ne 0\hfill \\ 0\hfill & \text{if}x=0\hfill \end{array}$

(a) Use the definition of derivative to compute f′(0).

(b) Show that f has derivatives of all orders that are defined on $ℝ$. [Hint: First show by induction that there is a polynomial p_n(x) and a nonnegative integer k_n such that $f^{(n)}(x)=p_n(x)f(x)/x^{k_n}$ for x ≠ 0.)