Stationary Points

1st Derivative Observations

• If the gradient, f '(a) = 0, it implies that there is a stationary point at x = a. 

• If the gradient, f '(x) changes from positive to zero to negative around x = a, there is a maximum point at x = a.

• If the gradient, f '(x) changes from negative to zero to positive around x = a, there is a minimum point at x = a.

• If the gradient, f '(x) changes from positive to zero to positive or negative to zero to negative around x = a, there is a stationary point of inflexion at x = a.

2nd Derivative Observations

• If the 2nd derivative, f ''(a) < 0 and f '(a) = 0, there is a maximum point at x = a.

• If the 2nd derivative, f ''(a) > 0 and f '(a) = 0, there is a minimum point at x = a.

• If the 2nd derivative, f ''(a) = 0, it may not be a point of inflexion. Its a necessary but insufficient condition.