Example for $\int_a^b f(x)g(x)dx\neq g(a)\int_a^c f(x)dx + g(b) \int_c^b f(x)dx$ for $f\geq 0$ and g a non-monotonic function.

Question

Well we all know about this theorem.

Let $f\geq 0$ be a function that is integrable over [a,b] and $g:[a,b]\to \mathbb{R}$ be a monotonic function. Then there is a $c \in [a,b]$ with $$\int_a^b f(x)g(x)dx= g(a)\int_a^c f(x)dx + g(b) \int_c^b f(x)dx$$.

My problem is, that I would really like to find a counter example to prove, that this theorem is in general not right when g is not monotonic. That means if $g(x_1)\nleq g(x_2) (g(x_1)\ngeq g(x_2))\forall x_1<x_2$

Is there anyone who could help me out. I would be very grateful.

Answer 1

Take a simple example where $g(a)=g(b)=0$ and $g(x)\ne 0$ in $(a,b)$. For example $g(x)=x^2-1$ in the interval $[-1,1]$, so $a=-1$, $b=1$. Use $f(x)=1$. Left hand side is the integral of a negative function, so it will be negative. Right hand side will be $0$.