Prove: there is a point where the derivative is zero

A small modification of your Mean Value Theorem for Integrals idea works.

By the MVT for integrals, there is a $c$ strictly between $-5$ and $-1$ such that the first integral is $4f(c)$.

Similarly, there is a $d$ between $-1$ and $3$ such that the second integral is $4f(d)$.

But the two integrals are equal, so $f(c)=f(d)$. The result now follows from Rolle's Theorem.


You need to show that $f$ has an extremum. If it doesn't then it must be strictly monotone and then the condition on integrals will fail (why?)

Tags:

Definite Integrals

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