# Reference for a property of convex function

Let $F:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function.

It is known that if $F$ has a strict local maximum, then it is not a convex function.

I just would like to ask you for a proper classic reference for that, I mean, a book where I can find that statement proved (or something similar).

ADDITION: I am just asking for a book where all those basic concepts involving convexity are dealt, no matter whether this very specific statement is proved or not.

EDITED: I replaced an absolute maximum with strict local maximum.

• I expect you will find many books proving that the graph of a convex function lies above any of its tangent planes; this result is a trivial consequence of that. And so I very much doubt that you will find a reference writing up this special case as a separate theorem – it is just too easy. – Harald Hanche-Olsen Jul 5 '14 at 9:39
• @Harald Hanche-Olsen, that is why I have edited my question to deal with strict local maximum. – Vicent Jul 5 '14 at 9:43

Suppose that $F$ has a strict local maximum at $x^*\in\mathbb R$. Then, there exists some $\varepsilon>0$ such that $f(x^*)>f(x)$ for all $x\in U$, where $U\equiv(x^*-\varepsilon,x^*+\varepsilon)\setminus\{x^*\}$.
To obtain a contradiction, suppose that $F$ is convex. Let $x_1\equiv x^*-\varepsilon/2$ and $x_2\equiv x^*+\varepsilon/2$. Clearly, $x_1,x_2\in U$ and $(x_1+x_2)/2=x^*$. By convexity and strict maximality of $x^*$, $$f(x^*)=f\left(\frac{x_1+x_2}{2}\right)\leq\frac{f(x_1)+f(x_2)}{2}\underbrace{<}_{!!!}\frac{f(x^*)+f(x^*)}{2}=f(x^*),$$ which is a contradiction.