Locate relative maxima, minima and saddle points of functions of two variables. Several examples with detailed solutions are presented. 3-Dimensional graphs of functions are shown to confirm the existence of these points. More on Optimization Problems with Functions of Two Variables in this web site.

b) If D > 0 and f

c) If D < 0, then f has a saddle point at (a,b).

d) If D = 0, then no conclusion can be drawn.

## Example 1Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by^{2} + 2xy + 2y^{2} - 6xSolution to Example 1:
Find the first partial derivatives f _{x} and f_{y}.
f _{x}(x,y) = 4x + 2y - 6
f _{y}(x,y) = 2x + 4y
The critical points satisfy the equations f _{x}(x,y) = 0 and f_{y}(x,y) = 0 simultaneously. Hence.
4x + 2y - 6 = 0 2x + 4y = 0 The above system of equations has one solution at the point (2,-1). We now need to find the second order partial derivatives f _{xx}(x,y), f_{yy}(x,y) and f_{xy}(x,y).
f _{xx}(x,y) = 4
f _{yy}(x,y) = 4
f _{xy}(x,y) = 2
We now need to find D defined above. D = f _{xx}(2,-1) f_{yy}(2,-1) - f_{xy}^{2}(2,-1) = ( 4 )( 4 ) - 2^{2} = 12
Since D is positive and f _{xx}(2,-1) is also positive, according to the above theorem function f has a local minimum at (2,-1).
The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6).
## Example 2Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by^{2} - 4xy + y^{4} + 2
A 3-Dimensional graph of function f shows that f has two local minima at (-1,-1,1) and (1,1,1) and one saddle point at (0,0,2).
## Example 3Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by^{4} - y^{4} + 4xy
A 3-Dimensional graph of function f shows that f has two local maxima at (-1,-1,2) and (1,1,2) and a saddle point at (0,0,0). ## ExercisesDetermine the critical points of the functions below and find out whether each point corresponds to a relative minimum, maximum, saddle point or no conclusion can be made.1. f(x , y) = x ^{2} + 3 y^{2} - 2 xy - 8x
2. f(x , y) = x ^{3} - 12 x + y^{3} + 3 y^{2} - 9y
## Answers to the Above Exercises1. relative maximum at (1,1) and (-1,-1) and a saddle point at (0,0) 2. relative maximum at (2,-3), relative minimum at (2,1), saddle points at (-2,-3) and (-2,1). ## More Links on Partial Derivatives and Multivariable FunctionsMultivariable FunctionsHome Page |