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)

Monotonic Function - dCode

Tag(s) : Functions

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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.

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

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

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