If I have a function of class $C^2$ and $f(-1)=f(0)=f(1)$ then I know there must be a $c \in {]-1,1[}$ such that $f'(c)=0$ by Rolle's theorem.
BUT is it also true that there must be a $c \in {]-1,1[}$ such that $f''(c)=0$ ?
Rolle theorem and 2nd derivative
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derivatives
1 Answers
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By Rolle's theorem, there must be a $c_1 \in \left]-1,0\right[$ such that $f'(c_1) = 0$, and there must be a $c_2\in \left]0,1\right[$ such that $f'(c_2) = 0$. If we define the function $g(x) = f'(x)$, we have $g(c_1) = g(c_2) = 0$ What does Rolle's theorem say about $g$?
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0Ah I see so can we say that if we have n equalities i.e f(a)=f(b)=f(c)=... ntimes in our interval then we must know that the ( n-1)th derivative must exist for $$ f^{n-1}(c)=0$$ ? – 2017-01-14
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1@SoHCahToha Yes, this approach is easily extended like that using indction. – 2017-01-14