Nonconvergence to f Consider the function

\(f(x)=\left\{\begin{array}{ll} e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right.\)

a. Use the definition of the derivative to show that \(f^{\prime}(0)=0\).

b. Assume the fact that \(f^{(k)}(0)=0\), for k = 1, 2, 3, .... (You can write a proof using the definition of the derivative.) Write the Taylor series for f centered at 0.

c. Explain why the Taylor series for f does not converge to f for \(x \neq 0\).

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