Suppose I have a differentiable and continuous function $f(x)>0$, the monotonicity of $f(x)$ is unknown.
Assume that $x_1< x_2< x_3 \in \mathcal{S}$, $\mathcal{S}$ is the domain of $f(x)$.
let $g(x)$ be \begin{equation} g(x_2)=2x_2-\frac{\int_{x_1}^{x_2}xf(x)dx}{\int_{x_1}^{x_2}f(x)dx}-\frac{\int_{x_2}^{x_3}xf(x)dx}{\int_{x_2}^{x_3}f(x)dx} \end{equation}
Actually, $g(x_2)$ is the $2x_2$ minus the sum of centroids in $(x_1,x_2)$ and in $(x_2,x_3)$.
My question is :
Is $g(x_2)$ is monotonically increasing as $x_2$ increases?
Some MatLab simulation results show that $g(x_2)$ seems to be a monotonically increasing of $x_2$. But, can we theoretically show this?
I have tried to see if the derivation of $g(x_2)$ is always large than $0$, but the result is not explicit.
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Thanks for the answer from @Greg Martin.
Actually, in my problem, $f(x)$ is an asymmetrical probability density function with $x\geq 0$.
Now, I still have a question: Does $g(x)=0$ have and only have one root?
Thanks for any helpful answers!
The answer is no. One idea for constructing a counterexample: suppose that $f(x)$ is an even function that is large at the endpoints of an interval $[-a,a]$ but small in the middle. Then when $x$ is not close to $a$, the centroid of the region under the graph of $y=f(x)$ on $[-a,x]$ will be very close to $-a$; but when $x$ approaches $a$, the centroid abruptly approaches $0$, at as high as a rate as we might like to arrange.
For a specific counterexample, take $f(x)=x^4$ and $(x_1,x_3) = (-1,2)$. In this case, $g(x)$ is decreasing from about $x=0.8$ to about $x=1.35$. (Here $f(x)$ is not strictly positive, but adding a tiny constant to $f(x)$ doesn't change $g(x)$ that much.)