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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A monotonic function is a function $ f $ such that for any $ x_1, x_2 $ if $ x_1 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

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Thanks to your feedback and relevant comments, dCode has developed the best 'Monotonic Function' tool, so feel free to write! Thank you!

Thanks to your feedback and relevant comments, dCode has developed the best 'Monotonic Function' tool, so feel free to write! Thank you!