# Differentiation: Mean Value Theorem

Let

r e-1/x2 i(x) = i 0

x740 x = 0

Show that the nth derivative of 1(x) exists for all n E N. Please justify all steps and be rigorous because it is an analysis problem. (Note: The problem falls under the chapter on Differentiability on IR in the section entitled The Mean Value Theorem.)

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#### Solution Preview

It is observed that: "The function g is non-zero, infinitely often differentiable, and any derivative of g at x = 0 equals zero." Now, how?:

This function is special, because when dealing with power series representations: if one wants to find a power series representation, one could apply Taylor's formula to find the coefficients of the power series, say, centered at zero. That requires that the function under consideration has to be infinitely often differentiable. This function is. Taylor's formula involves derivatives of the function at the origin. In this case, they are all zero. Hence, the power series associated with this function would be identically equal to zero. But then it does not represent the original function.

In other words, there are functions for which you can use Taylor's theorem to find a convergent power series, ...

#### Solution Summary

The existence of an nth derivative is proven.