I have read in following paper "OPTIMIZATION FOR PRODUCTS OF CONCAVE FUNCTIONS" that if $f_1(x)$ $f_2(x)\cdots f_n(x)$ are positive concave functions in some interval $[a,b]$ then their product $h(x)=f_1(x)f_2(x)\cdots f_n(x)$ has following property.
1- There exist points $\alpha$ and $\beta$ with $a\leq \alpha \leq\beta\leq b$ such that $h$ is strictly increasing on $[a,\alpha)$, constant on $(\alpha,\beta)$ and strictly decreasing on $(\beta,b]$.
In my thinking this means that $h(x)$ can not have isolated maxima in $[a,b]$ (due to the conditions described for the derivative). However I have seen some examples where the function can have more than one maxima and the sign of the derivative also changes more than once. Can anybody please explain me where I am wrong in understanding this.
The first function first increases up to the unique global maximum on $[0,1]$ and then it decreases. It does not have multiple maxima that are isolated.
For the second example $h(x)=(1-x)(1-x^2)$, the unique global maximum on $[0,1]$ is $0$ and it is a decreasing function on $[0,1]$.
Edit: I have included the graph of the first function.