How do I find the nature of a stationary point

Firstly, note that a stationary point can either be a local maximum, local minimum or point of inflexion. Now suppose you have a differentiable function (a graph which has a unique gradient/derivative at each point), say f(x) which has a stationary point at x = a. Then we will have that the derivative at that point is f'(a) = 0. Now let f''(a) denote the 2nd derivative at a. Then, f''(a) > 0 implies that there must be a local minimum at a whilst f''(a) < 0 implies that there must be a local maximum at a. These don't need to be memorized (phew!) - if you draw a generic graph of a local maximum and local minimum you can figure out which way round the signs need to be (I would proceed to draw these graphs and explain them). However, if f''(a) = 0 then the test is inconclusive so we resort to a 2nd method. 2nd method: You can calculate the signs of the derivatives either side of x = a and close to a. For example if f'(x) < 0 for x < a and f'(x) > 0 for x > a then we have a local minimum (again a graph is very useful here).