I came across this question from a competitive exam:
$\textbf{Problem: }$Given $f:[\frac{1}{2},2]\to\mathbb{R}$ a strictly increasing function, define $g:[1,2]\to\mathbb{R}$ by $g(x)=f(x)+f(\frac{1}{x})$. Does there exist a suitable $f$ for which there is a partition $P$ of $[1,2]$ such that $U(P,g)=L(P,g)$? Here, $U(P,g)$ and $L(P,g)$ are the upper and lower Riemann sums respectively.
I could figure this out and the answer is $\textbf{Yes}$. One can take $f(x)=\log x$, $f(x)=\frac{x-1}{x+1}$, etc. So this is alright.
I thought of making $g$ to be a constant function to construct the above examples. While trying to come up with these examples, I made some rough graphs that satisfied the hypotheses of the problem and observed something that I want to ask about. Later, I drew several graphs in Desmos as well and observed the same thing. My question is, is the following true?
$\textbf{Question:}$ For $a>1$, given $f:[\frac{1}{a},a]\to\mathbb{R}$ a monotonic function, define $g:[1,a]\to\mathbb{R}$ by $g(x)=f(x)+f(\frac{1}{x})$. Is $g$ always non-decreasing?
I feel it is true. But I could not prove it. Neither could I construct a counterexample. Any hints to prove/disprove the above statement are appreciated.
If the above result is true, then the only possible candidates for $g$ in the above original problem can be constants, right? That is why I got interested in this.
$\textbf{Please note:}$ I am not assuming any continuity conditions anywhere. Just monotonicity.
$f(x) = -x$ and $a=2$ is a counterexample to the question. Moreover, $g(x)$ is not necessarily monotonic if $f(x)$ is. This is due to the counterexample $f(x) = x^2-4.5x+3.5$ and $a=2$. $f(x)$ is monotonic but $g(x)$ switches from decreasing to increasing at around $1.64$, which is a root of multiplicity $1$ of $g'(x)$.