Monotonicity at a Point - How is it checked?

Question

As far as I know, for a given function f(x) if we have to check monotonicity at a point x = a, we check which ones of the following inequalities (or equalities) are being satisfied -

1. f(a + h) > f(a) > f(a - h)
2. f(a + h) < f(a) < f(a - h)
3. f(a + h) > f(a) = f(a - h)
4. f(a + h) = f(a) > f(a - h)
5. f(a + h) < f(a) = f(a - h)
6. f(a + h) = f(a) < f(a - h)
7. f(a + h) > f(a) < f(a - h)
8. f(a + h) < f(a) > f(a - h)
9. f(a + h) = f(a) = f(a - h) for small h

My confusion lies here. Does the value of the function's derivative at x=a have anything to do with its monotonicity at x=a?

For example, let's discuss the monotonicity of y = x² at x = 0. Using the definitions stated above, it is easy to observe that f(0+h) > f(0) and f(0-h) > f(0) but f'(0) = 0.

Could someone please help clear the confusion, and tell what's the real definition for monotonicity at a point and also why the two above methods don't agree with each other?

Thanks a lot!

Answer 1

It does tell you the monotonicity when the derivative is non-zero. When it is zero, it's a critical point, so we can't determine if it's monotone or not.

In your example, you're totally correct, it's inconclusive since for $f(x) = x^2$, $f'(0) = 0$, and it's not monotone. However $g(x) = x^3$, $g'(0) = 0$ but it's monotone.

So the idea is that for a continuous $f(x)$ that's differentiable on an interval containing $a$ (please correct me if this is not the exact condition), $f'(a) \neq 0 \implies $ $f(x)$ monotone around $x=a$

Answer 2

We only talk about monotonicity on an interval — not a point.

So the answer to your question is: they are not necessarily related. If the function is infinitely differentiable (such as polynomials, exponentials, etc), then $f'(x_0)>0$ implies that $f'$ is positive on an interval containing $0$. Then the mean value theorem shows you that $f$ is increasing on that interval.

However, this site gives a function which has positive derivative at $0$ but which is not increasing on any interval containing $0$.

Answer 3

To answer your question in short, yes. The derivatives of the function does tell us about the monotonicity of the function. But when $f'(x)=0$, its a critical point (i.e. the function is not differentiable or the slope is zero)

In such cases, we check the inequality around the point. In this case for $f(x)=x^2$, $f'(0)=0$

Here $f(a-h) > f(a) < f(a+h)$, since its neither increasing nor decreasing at the given point, we can say that its the point of extrema. This can also be verified from the graph of $y=x^2$.

So we can't state that the function is monotonus in its natural domain. But if we change the domain to $x ∈ (0,∞)$ or $x ∈ (-∞,0)$, we can say the function is monotonically increasing or monotonically decreasing respectively.

Hope it helps.