Could someone give me some help with the following question? I'm not sure how to tackle this question because it involves a piecewise function. Any help is appreciated!

Consider the function f: [-1,1] defined by
$f(x)=(x+1)^2+e^{-1/x^2}, x≠0$
$f(x)=1,\ x=0$
(a) Establish whether f satisfies the hypotheses of the Mean Value Theorem on the given interval
(b) Regardless of the answer to part (a) show that f satisfies the conclusion of the Mean Value Theorem with c=0. Be aware that this doesn't tell us anything about part (a)!

  • $\begingroup$ You need to determine if $f$ remains differentiable on the interval $(-1, 1)$. Is $f$ differentiable at $0$? You'll need to take the derivative at $x=0$ explicitly to find out (which after simplification becomes $\lim_{h \to 0}\frac{h^2 + 2h + e^{-1/h^2}}{h}$). $\endgroup$ – Chris Oct 14 '17 at 6:15

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