Another sufficient condition of local minima/maxima Let a function have continuous second order partial derivatives at a point and let If for all , then has a local maximum at A. If for all , then has a local minimum at A.

Relative maximum Let be defined on a set and let a set U be defined by the constraints If, for a point , there is a neighbourhood such that , we say that has a local maximum at A relative to U.

Relative minimum Let be defined on a set and let a set U be defined by the constraints If, for a point , there is a neighbourhood such that , we say that has a local minimum at A relative to U.

Method of Lagrange multipliers Let U be given by the set of constraints We are to find local minima or maxima of relative to U. First we define a so-called Lagrange function as The numbers are called Lagrange multipliers.

We are now looking for points where the first partial derivatives of are equal to zero (these are sometimes called stationary points) and at the same time satisfy the constraints. This leads to the following system of equations:

We have n + k equations for n + k unknowns By solving the system , we find a set of candidates for local maxima/minima.

Finally, we test each whether using the fact that the first differentials of the constraints have to be zero at X:

Examples 1 Constraint: 2 Constraints:

Results 1 Relative local maximum at [1.5,-0.5] 2 Relative local minimum at [0,-1,2]