4

Let $f:[a, b] \rightarrow \mathbb{R}$ where $a, b \in \mathbb{R}$ with $a <b$. Suppose $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose $f$ is monotonically increasing on $[a,b]$. Show that $f'(x) \ge 0$ for all $x \in (a,b)$.

My attempt: I tried using the Mean Value Theorem, but it doesn't quite seem to work. For example, by the MVT we can conclude that there exists a $c \in (a,b)$ such that $f(b) - f(a) = f'(c) (b-a)$. Which implies that $f'(c) = \frac{f(b) - f(a)}{b-a}$. Now since $f$ is monotonically increasing, $f(b) - f(a) \ge 0$ whenever $b>a$, so $f'(c) \ge 0$. But this only shows for one particular $c \in (a,b)$, and the question asks to show this is true for ALL $x \in (a,b)$. What can I do to complete the proof?

mdcq
  • 1,658
user40333
  • 985
  • 3
    My attempt: I tried using the Mean Value Theorem, but it doesn't quite seem to work. May I ask, then, why you chose to put "using mean value theorem" in the title? Maybe MVT is indeed not the right way to solve this, and such a title can only discourage better alternative answers. – dxiv Aug 08 '17 at 03:52

4 Answers4

9

For $x \in (a,b)$ $$f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}\geq0$$

Sahiba Arora
  • 10,847
5

Your approach using MVT will never guarantee the location of $c$ or may not cover all $c\in(a, b)$. Consider for example $f(x) = x^{3}$ in $[-1,1]$. Since $f$ is strictly increasing it follows that the ratio $(f(b) - f(a)) /(b-a) >0$ for any two distinct points $a, b\in[-1,1]$ and thus the corresponding $c$ guaranteed by MVT will always have $f'(c) >0$. This misses the point $c=0$ where derivative vanishes. Thus your proof can not be salvaged. Even isolating such troublesome points like the one given in above example and then showing that derivative vanishes there is a bit difficult.

The proper and much simpler approach is to use the definition of derivative as given in Sahiba Arora's answer.

Paramanand Singh
  • 87,309
0

Hint: For a given $x$, there exists $a \le c \le x \le d \le b$ such that $$\frac{f(d)-f(c)}{d-c}=f'(x).$$

Math Lover
  • 15,153
  • But that isn't what the Mean Value Theorem itself says. – John Coleman Aug 08 '17 at 01:59
  • @JohnColeman The idea is to consider a 'new' function defined on $(c,d)=(x-\delta,x+\eta)$ such that the mean-value theorem gives the desired result. – Math Lover Aug 08 '17 at 02:03
  • How would you finish your proof @Math Lover? – user40333 Aug 08 '17 at 02:09
  • @user40333 For any given $x \in (a,b)$, we consider the function defined on $[y,z]$, where $x \in (y,z)$, and $a \le y < x < z \le b$ such that $\frac{f(z)-f(y)}{z-y}=f'(x)$. Since $f(z) \ge f(y)$, $f'(x) \ge 0$. – Math Lover Aug 08 '17 at 02:13
  • 3
    The problem is that unless I am missing something your statement is (in general) false. The converse of the Mean Value Theorem doesn't hold without some additional assumptions. For example $f(x) = x^3$ satisfies $f'(0) = 0$ but $\frac{f(d)-f(c)}{d-c}$ never equals $0$ for any $c,d$ which straddle $0$. Now it might be the case that $f$ increasing is enough to guarantee the result, but if so, that would be a result which is harder to show than the result being proved. – John Coleman Aug 08 '17 at 02:22
  • Nice comment @JohnColeman. I really like you comment. I believe that if the function is monotonically increasing or decreasing then we should be able to find an interval straddling $x$. – Math Lover Aug 08 '17 at 02:28
  • It probably isn't necessary to have equality, if we can assume that $f'(x)$ is continuous. In that case, I see how to tweak your answer into a proof, but continuouty isn't assumed in the problem statement. Perhaps something like Darboux's Theorem is relevant, but then all of a sudden you are talking about a non-trivial proof. This problem is a lot more interesting than I thought at first. – John Coleman Aug 08 '17 at 02:34
  • @JohnColeman Yes, indeed it is. – Math Lover Aug 08 '17 at 02:44
  • @JohnColeman it might be the case that f increasing is enough to guarantee the result Your (counter)example $f(x)=x^3$ is increasing, in fact strictly increasing. – dxiv Aug 08 '17 at 02:47
  • Fully agree with @dxiv. The function $f(x) =x^{3}$ is strictly increasing in any interval. This has nothing to do with $x=0$ being a straddle point. Perhaps you are using a different (uncommon/non-popular) definition of monotone function. – Paramanand Singh Aug 08 '17 at 02:50
  • Sorry, I realized my mistake. – Math Lover Aug 08 '17 at 02:50
  • @dxiv You are right of course. When I typed that I was thinking about the derivative of $f$ rather than $f$ itself. Thus this answer is difficult to salvage. – John Coleman Aug 08 '17 at 02:51
  • @dxiv Can we apply this answer for those $x$ where $f'(x)>0$ (or $f'(x) <0$ for monotonically decreasing functions)? I think so. Any thoughts? – Math Lover Aug 08 '17 at 03:02
  • @MathLover No, consider for example $,f(x)=x^3+x,$ at $,x=0,$. One sufficient condition would be that $,f,$ doesn't have an inflection point at the particular $,x,$ you are considering, see A converse proposition to the Mean Value Theorem for example (though that and the linked question assume twice differentiability). – dxiv Aug 08 '17 at 03:35
  • Thanks a lot @dxiv. – Math Lover Aug 08 '17 at 04:00
0

Your proof can be made to work if $f'$ is continuous. Let $x∈ (a,b)$. Then for each $n\gg 1$, apply MVT as you have on the interval $(x-\frac1n,x+\frac1n)$ to discover an $x_n$ with $|x_n - x|<1/n$ and $f'(x_n)\geq 0$. Just take $n→∞$ to conclude, using continuity of $f'$.

Calvin Khor
  • 34,903