Convexity and local maxima If a continuous function $f$ on $(a,b)$ is not convex, there is some choice of number $m$ so that $g(x)=f(x)+mx$ has a local maximum at a point $z$ inside the interval $(a,b)$. Is that true?
 A: A local maximum, yes.  There exist $c,d$ such that $f((c+d)/2) > f(c)/2 + f(d)/2$.  Adjust $m$ so that $g(c) = g(d)$...
A: $\newcommand{\Reals}{\mathbf{R}}$Referring to your earlier closely related (now deleted) question, suppose $f:(a, b) \to \Reals$ is a continuous function. Let's say $f$ has:


*

*A peak if there exist numbers $x < z < y$ in $(a, b)$ such that $\max\bigl(f(x), f(y)\bigr) \leq f(z)$.

*A valley if there exist numbers $x < z < y$ in $(a, b)$ such that $f(z) \leq \min\bigl(f(x), f(y)\bigr)$.
If $f$ has a peak, then with the preceding notation, an easy application of the extreme value theorem on $[x, y]$ shows that $f$ has a (possibly non-strict) local maximum in $(x, y)$, and hence in $(a, b)$. Conversely, if $f$ has a local maximum at a point $z$ of $(a, b)$, then $f$ obviously has a peak.
An entirely analogous argument shows that $f$ has a valley if and only if $f$ has a local minimum in $(a, b)$.
Particularly, $f$ has a local extremum (maximum or minimum) if and only if $f$ is not strictly monotone. 
