Lecture 10 Function f and trying to nd extrema Find max and min points Find local max and mins Three places to look for extrema Point where the derivative is zero Point where the derivative is not de ned Point on the boundary Critical points points where the derivative is zero or not de ned Second derivative test for functions with one variable There is a test that can tell us local max or local min Quadratic approximation Second derivative test for functions with two variables We know that f f 0 at the critical point Saddle point x y critical point at 0 0 Saddle point x y critical point at 0 0 Example Easy case B 0 Minimum if A and C are positive Maximum of A and C are negative Saddle point if A and C are different signs Now say that B 0 try completing the square Minimum if A and AC B are positive Maximum if A is negative and AC B is postive Saddle if AC B is negative Degenerate case if AC B is zero Quadratic approximation Now f x y f f 0 A f B f C f Example 13 10 5 f 2x 2xy y 4x 2y 1 f 4x 2y 4 and f 2x 2y 2 with a critical point at 3 4 A f 4 B f 2 C f 2 f 3x x 3xy f 3 3x 3y and f 6xy with critical points at 1 0 0 1 1 0 0 1 Example saddle point
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