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Monotonic Function

Tool to calculate the monotonicity (or not) of a function, i.e. check its direction of variation, if a function is (strictly?) monotonic (increasing or decreasing)

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Monotonic Function -

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# Monotonic Function

## Monotonic Function Tester

### What is a monotonic function?

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.

### How to determine if a function is monotone?

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

### What is a monotonic sequence?

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

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