Question

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for `f(x) = x² + 2x - 3`?

Answer 1

Use the derivative. The function is (strictly) increasing over intervals where `f'(x) > 0` and (strictly) decreasing over intervals where `f'(x) < 0`. Local extreme points occur at critical points where `f'(x) = 0` or where `f'(x)` is undefined.

Explanation:

If `f(x)=x^2+2x-3`, then `f'(x)=2x+2` so that `f'(x) < 0` when `x < -1` and `f'(x) > 0` when `x > -1`. This means `f` is decreasing over the interval `x <= -1` and `f` is increasing over the interval `x >= -1`. `f` has a local minimum value at the critical point `x=-1` equal to `f(-1)=1-2-3=-4`.