How can I prove that $f(x)$ is convex on an interval if and only if $f'(x)$ is monotonically increasing ?

• Let $\lambda \in (0,1)$ and $x,y \in I$. $f(x)$ is convex on an interval, if the statement $f((1-\lambda)x + \lambda y) \leq (1-\lambda)f(x) + \lambda f(y) $ is true for all $x,y$.

• $f'(x)$ is monotonically increasing on an interval $I$, if $0 \leq \frac{d f'(x)}{dx}$ is true for all $x \in I$.

I don't have a problem understand this, with a graph it's quite easy to understand. But I can't manage to prove it formally, can someone give me a hint?

Note that I do have seen this question, but since I can't use inequality proven previously it doesn't help me.


Hint: Convexity of $f$ can be expressed as $$a < x < b \in I \implies f (x) \leq f(a) + \frac{f(b) - f(a)}{b -a} (x - a)$$

or $$a < x < b \in I \implies f (x) \leq f(b) + \frac{f(b) - f(a)}{b -a} (x - b)$$

Then $f$ is convex in $I$ if, and only if,

$$a < x < b \in I \implies \frac{f (x) - f(a)}{x-a}\leq \frac{f(b) - f(a)}{b -a} \leq \frac{f(b) - f(x)}{b -x}$$

In fact, one needs only one of those inequalities to characterize $f$ convex.

Edit: Let $x \to a$ on the first inequality and $x \to b $ on the second inequality then

$$\begin{align} a < b \implies f'(a) &\leq \frac{f(b) - f(a)}{b -a} \leq f' (b) \\a < b &\implies f'(a) \leq f'(b) \end{align}$$ Can you take it from here?

  • $\begingroup$ I understand these inequalities, but I don't know how to conclude that f' has to be monotonically increasing $\endgroup$ – Christian Apr 15 '15 at 16:57
  • $\begingroup$ One more edit then $\endgroup$ – Aaron Maroja Apr 15 '15 at 16:58
  • $\begingroup$ Sorry but I don't see how $f'(a) \leq f'(b) $ helps me $\endgroup$ – Christian Apr 15 '15 at 17:29
  • $\begingroup$ Transitivity of $\leq$, that is: if $a \leq b$ and $b \leq c$ then $a \leq c$. And $$f'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a}$$ also $\lim_{x \to a} c = c$ ,where $c$ is a constant. $\endgroup$ – Aaron Maroja Apr 15 '15 at 17:31
  • $\begingroup$ But how can I use this to prove that $f'$ has to be monotonically increasing? $\endgroup$ – Christian Apr 15 '15 at 18:43

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