How do I find and determine the nature of stationary points of a function?

There are 3 types of stationary points for functions: a maxima, a minima and a saddle point. They all occur when the derivative of a function, f(x), is equal to 0, e.g. f'(x)=0. Therefore in order to find the stationary points we must differentiate our function, set the derivative equal to 0 and solve for x. Then we take our value for x and substitute back into our original function to find the y coordinate. e.g.f(x) = 2x^3 - 6x + 5f'(x) = 6x^2 - 6 = 0 -> x = +1,-1 -> y = +1, +9 -> stationary points at (1,1) and (-1,9).In order to determine the nature of the stationary point we must use the second derivative of the function, f''(x). Then, subbing our value for x at the stationary point into f''(x) will give 1 of 3 options: f''(x) > 0, f''(x) < 0 or f''(x) = 0. A negative value corresponds to a local maxima, a positive value corresponds to a local minima and a null value means further testing is required. Taking a point either side of our stationary point and repeating the first derivative test can tell us if we have a maxima, a minima or a saddle point.e.g. f''(x) = 12x -> sub (1,1) -> f''(x) = 12 therefore local minima -> sub (-1,9) -> f''(x) = -12 therefore local maxima.