1. 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.

2. 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.

3. 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.

4. 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.

5. 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:

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

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

8. Examples 1
Constraint: 2
Constraints:

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