Critical point

How do we find extreme points of functions of one variable?

Let y = f(x). To find the local maximum or minimum, proceed as follows:

 - compute f′(x),

 - solve f′(x) = 0 for x,    
   (also find x's for which f′(x) does not exist)

 - these are the critical points of f.

 - some of these are maxima, some are minima, others are neither (in this case, they will be called "saddles").

 - 2nd derivative test: classify the critical points based on the sign of f(a).
   Note that it is possible that a is a minimum and f′′(a) = 0. Example: y = x4 at a = 0.

 - 1st derivative test: classify the critical points based on change of the sign of f'(a)
   f′(x) > 0 for x < a and f′(x) < 0 for x > a,
   f′(x) < 0 for x < a and f′(x) > 0 for x > a,
   no change.