Question

How do you know a function is increasing?

Answer 1

It will be increasing when the first derivative is positive.

Explanation:

Take the example of the function `f(x) = e^(x^2 - 1)`.

The first derivative is given by `f'(x) = 2xe^(x^2 - 1)` (chain rule). We see that the derivative will go from increasing to decreasing or vice versa when `f'(x) = 0`, or when `x= 0`.

Whenever you have a positive value of `x`, the derivative will be positive, therefore the function will be increasing on `{x|x> 0, x in RR}`.

The graph confirms

Hopefully this helps!