Monotonic Function

Question 1

What is a monotonic function? (Definition)

Answer

A monotonic function is a function $ f $ such that for any $ x_1, x_2 $ if $ x_1 < x_2 $ then either $ f(x_1) < f(x_2) $ (increasing function) or $ f(x_1) > f(x_2) $ (decreasing function) but not both.

In other words, a monotonic function is a function which preserves or reverses the order.

Question 2

How to determine if a function is monotone?

Answer

Several methods are used to calculate the direction of variation of a function in order to know if a function is monotonic:

— Calculation with its derivative: When the derivative of the function is always less than $ 0 $ or always greater than $ 0 $ then the function is monotonic.

Example: The derivative of the function $ f(x) = x^3+1 $ is $ f'(x) = 3x^2 $, the calculation of $ f'(x) >= 0 $ is always verified, whatever the value of $ x $ so the function $ f $ is monotonic (and increasing) on $ \mathbb{R} $

— Calculation with its equation: Certain functions are notoriously monotonic (either increasing or decreasing): the exponential function, etc.

Example: $ x+1 $ is increasing over $ \mathbb{R} $

— Calculation with the curve of the function: a monotonic function has its curve which goes either always downwards or always upwards

Question 3

What is a monotonic sequence?

Answer

A monotonic sequence is a series of numbers either always increasing or always decreasing.

Monotonic Function