Increasing and monotonically increasing: related to the first derivative

[songoku]

TL;DR

What is the condition required (related to first derivative) for a function to be increasing function and monotonic increasing function?

I will start from the meaning of increasing function. A function is said to be increasing function if for x < y then f(x) ≤ f(y). Is this correct?

Then f(x) is increasing function if f'(x) ≥ 0. Is this correct?

Lately I encounter the term "monotonic increasing". What is the difference between monotonic increasing and increasing? Is monotonic increasing function the one where if x < y then f(x) < f(y)?

Then f(x) is monotonic increasing function if f'(x) > 0?

 

[Infrared]

The key is the mean value theorem.

Suppose $f$ is a differentiable function defined on an interval and $f'(x)>0$ for all $x$. The mean value theorem says that if $a<b$, then $f(b)-f(a)=f'(t)(b-a)$ for some $t\in (a,b)$. The righthand side is positive, so $f(a)<f(b)$.

Similarly, $f'(x)\geq 0$ will imply $f(a)\leq f(b)$.

To distinguish these two cases unambiguously, I would use the terminology "strictly increasing" and "weakly increasing" (or maybe "non-decreasing").

To use the mean value theorem, you needed to assume that the domain of $f$ is an interval. This is important. Think about the function $f(x)=-1/x$. Then $f'(x)=1/x^2>0$ for all nonzero $x$, but $f(-1)=1>-1=f(1)$.

 

[andrewkirk]

Then f(x) is monotonic increasing function if f'(x) > 0?

You need to add "and f is differentiable everywhere". Otherwise, consider the function f(x)=x for x<0 and f(x)=x-1 for x>=0. This is not differentiable at x=0 and decreases there, so it is not an increasing function.

The word 'monotonic' adds no information to the word 'increasing'. Monotonic functions can be increasing or decreasing. All increasing and decreasing functions are monotonic. Every monotonic function is either increasing or decreasing.

It is common to use the word 'strictly' to indicate replacement of >= by =. That is, f is increasing if for all x and y, x>y --> f(x)>=f(y), and f is strictly increasing if for all x and y, x>y --> f(x)>f(y).

 

[PeroK]

Summary:: What is the condition required (related to first derivative) for a function to be increasing function and monotonic increasing function?

I will start from the meaning of increasing function. A function is said to be increasing function if for x < y then f(x) ≤ f(y). Is this correct?

Then f(x) is increasing function if f'(x) ≥ 0. Is this correct?

Lately I encounter the term "monotonic increasing". What is the difference between monotonic increasing and increasing? Is monotonic increasing function the one where if x < y then f(x) < f(y)?

Then f(x) is monotonic increasing function if f'(x) > 0?

Note that the function $f(x) = x^2$ is strictly increasing for $x \ge 0$, yet $f'(0) = 0$. Or, perhaps a better example is that $x^3$ is strictly increasing for all $x$, yet $f'(0) = f''(0) = 0$.

There was another example came up in a homework problem recently:

$f(x) = \frac 1 2 x^2 - x\cos x + \sin x$

$f$ is strictly increasing for $x \ge 0$, but has a sequence of points at which $f'(x) = 0$.

 

[mathwonk]

to answer your specific questions, yes f' ≥ 0 everywhere on an interval (note it is required that the domain be a connected interval, or else counterexamples exist) does imply increasing in your sense, and f'>0 everywhere on an interval does imply strictly increasing on that interval. As noted, this is not required, since it is possible to have a strictly increasing function which is differentiable everywhere on an interval, and yet the derivative is zero at some points, as in post #4.

 

[mathman]

Summary:: What is the condition required (related to first derivative) for a function to be increasing function and monotonic increasing function?

I will start from the meaning of increasing function. A function is said to be increasing function if for x < y then f(x) ≤ f(y). Is this correct?

Then f(x) is increasing function if f'(x) ≥ 0. Is this correct?

Lately I encounter the term "monotonic increasing". What is the difference between monotonic increasing and increasing? Is monotonic increasing function the one where if x < y then f(x) < f(y)?

Then f(x) is monotonic increasing function if f'(x) > 0?

You got the definitions reversed. Monotone increasing means $f(x)\le f(y)$ for $x\lt y$. (Strictly) increasing if $f(x)\lt f(y)$.

 

[songoku]

Thank you very much for all the explanation

 

[mathwonk]

note i corrected a misprint in my post from second to first derivative.