Show that a function (the F-measure) of two monotonic functions (one increasing, one decreasing) has at most one maximum This is a pure maths question that emerges from information retrieval. 
The performance of an algorithm is typically measured with Precision and Recall, functions that vary between $0$ and $1$. These can then be combined into a single measure of effectiveness with a variable, $\beta$, that determines how much weight should be given to Precision and how much to Recall. This is called the $F_\beta$-measure, also varies between $0$ and $1$, and the formula is:
$$F_\beta = \frac{1 + \beta^2}{\beta^2 P + R} PR$$.
Typically, the algorithm developer can adjust the confidence threshold (also on the scale between $0$ and $1$) and generate readings for both Precision and Recall at each level. This will yield characteristic curves for Precision and Recall. One of the characteristics is that, as the threshold is raised, Precision is monotonically increasing and Recall is monotonically decreasing.
The question then is this:
Assuming that Precision is monotonically increasing and Recall is monotonically decreasing, can we prove that $F_\beta$ has at most one maximum?
Here is a typical chart of Precision, Recall and $F_\beta$ for a variety of values of $\beta$ where the y-axis shows the value of the function and the x-axis shows the confidence threshold:
Click for chart of $F_\beta$ for various $\beta$ and Precision and Recall
 A: The reciprocal of $F$ is a weighted sum of $1/R$ and $1/P$:$$\frac 1 F = \frac {\beta^2}{1+\beta^2} \frac 1 R + \frac 1 {1+\beta^2} \frac 1 P,$$ that is, a convex combination of an increasing function and a decreasing function.
In general, convex combinations of increasing and decreasing functions are not convex, not concave, and can have multiple local maxima and minima. The best that can be said of them in general is that they are of bounded variation. 
For example (with $\beta=1$), suppose on a small neighborhood of $1/2$, $1/R(x) = x+\sin(Nx)/N^2$ (which is monotone increasing, and $1/P(x) = 1-x$ (which is monotone decreasing), when $N>1.$ On that neighborhood, $1/F(x) = \sin(Nx)/2N^2$, which (if $N$ is big enough) has multiple local maxima and minima.  So will $F$.
If you added the hypotheses that $1/R$ and $1/P$ were convex functions, then $1/F$ would be, too, and have (typically) a unique minimum, so $F$ would have a unique maximum.  I say typically because the minimum of $1/F$ (and maximum of $F$) might be attained on an interval.  I have no idea if adding this convexity hypothesis is reasonable in the field of "information retrieval", but it does lead to a simple and understandable proof of your desired conclusion.
