# Sum of a concave and convex function: how to show the endpoints are global extremas

EDIT

To clarify: I am working with specific $$f$$ and $$g$$. They are algebraically very tedious, which makes solving for FOC and SOC rather difficult. I believe the global extremum of my $$f+g$$ exists only at the endpoints and I'm asking what conditions do I need to check to prove that. Hypothetical counter-examples aren't that helpful because my $$f$$ and $$g$$ are specific.

Thanks!!

Original Post

Suppose I have two functions $$f$$ and $$g$$, both on [0,1]. Suppose I know $$f$$ is strictly decreasing and convex. $$g$$ is strictly increasing and concave. And both functions are negative on [0,1]. Is there anything I'd know about $$f+g$$?

I'm basically trying to show that the global maximum of $$f+g$$ lies always on 0 and/or 1(the endpoints). Any clues on how to show that or what would I need to show that would be greatly appreciated!

P.S. I do expressions of these functions but they are algebraically very tedious. Meaning it's very difficult to literally "solve" for anything but I suppose I could check for certain properties.

Thanks!!

With only the given properties of $$f$$ and $$g$$, there need not be a global maximum at $$0$$ or $$1$$. To show this, let $$f(x)=-2x$$ and $$g(x)=-x^2+3x-2$$. It is easy to see that these functions meet your criteria and that $$f+g\,$$ has a unique maximum at $$x=\frac{1}{2}$$. A small modification will even allow $$f\,$$ to be strictly convex.
Note that the condition that both $$f$$ and $$g$$ be negative is irrelevant: Since both $$f$$ and $$g$$ are bounded, you can always add a suitable constant to make both $$f$$ and $$g$$ negative, because all other mentioned properties are invariant under addition of a constant, in particular the location of the maxima.
• Thanks overglow! The properties I mentioned aren't necessarily the only given properties of $f$ and $g$. As said, I do have explicit expressions of $f$ and $g$ and I'm basically asking what other conditions do I need to verify to prove that the maximum locates at the endpoints. Jun 24, 2021 at 9:45