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Theorem
Let f be a function with two variables with continuous second order partial derivatives f_{xx}, f_{yy} and f_{xy} at a critical point (a,b). Let

D = f_{xx}(a,b) f_{yy}(a,b) - f_{xy}^{2}(a,b)

If D > 0 and f_{xx}(a,b) > 0, then f has a relative minimum at (a,b).

If D > 0 and f_{xx}(a,b) < 0, then f has a relative maximum at (a,b).

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

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

We now present several examples with detailed solutions on how to locate
relative minima, maxima and saddle points of functions of two
variables. When too many critical points are found, the use of a table
is very convenient.

=-2x+10

=3y^2-6y-9

x=5

y=9 or 2

minima= (5, 2)

Please let me know if you need any clarification. I'm always happy to answer your questions.

Jul 22nd, 2015

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