Math 241 Spring 2014 Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Math 241 Spring 2014 Local minima Definition f Rn R has an local minimum at the point p if there is a 0 such that f p f x for 0 kx pk Math 241 Spring 2014 Critical Points Idea If any component of the gradient vector is non zero the function will change increase decrease in that direction So local maxima and minima of a function f happen at places where f 0 Math 241 Spring 2014 Critical Points Definition The critical points of the function f are exactly the places where f 0 Question How can we tell when a critical point is a minimum or a maximum Math 241 Spring 2014 Saddle Points In higher dimensions other things can happen Math 241 Spring 2014 Potato Chips http freakonomics com 2011 04 15 the math of pringles Math 241 Spring 2014 Second derivative test 2 variables Like in 1 variable we have an second derivative test For f f x y at a critical point a b we consider Second partials fxx a b fyy a b fxy a b fyx a b Discriminant D fxx a b fyy a b fxy a b 2 Math 241 Spring 2014 Second derivative test 2 variables Theorem Let a b be a critical point for the function f R2 R i e f a b 0 If D 0 fxx a b 0 f a b is a local minimum for f D 0 fxx a b 0 f a b is a local maximum for f D 0 f a b is neither a local maximum or minimum for f Math 241 Spring 2014 Examples Math 241 Spring 2014 Absolute Extrema Closed and Bounded Sets Definition A set A Rn is closed if it contains all of its boundary points Definition A set A Rn is bounded if it can be contained in a ball of finite radius Math 241 Spring 2014 Absolute Extrema Extrema Theorem If f is a continuous function it attains an absolute maximum and an absolute minimum on the set A Math 241 Spring 2014 Absolute Extrema Looking for extrema Look at critical points look at boundary Question How to deal with the boundary Math 241 Spring 2014 Constrained Optimization and Lagrange Multipliers 14 8 Constrained Optimization Idea Often we re interested in optimizing one function f x y on a region defined by another function g x y That is we want to find extrema of f on the region A x y g x y L Inside the region we look for critical points on the boundary we need to do something different Math 241 Spring 2014 Constrained Optimization and Lagrange Multipliers 14 8 Constrained Optimization Question How do we optimize the function f x y along the level curve x y g x y L Math 241 Spring 2014 Constrained Optimization and Lagrange Multipliers 14 8 Gradients Idea We want to find points on the level curve x y g x y L where the function f doesn t change in the direction of the tangent line to the curve That is we want to find x0 y0 where the tangent vector v to the level curve has Dv f x0 y0 0 Math 241 Spring 2014 Constrained Optimization and Lagrange Multipliers 14 8 Normal Vectors Idea If we want Dv f x0 y0 0 where v is the tangent vector to the level curve for g we need the gradient f x0 y0 to be perpendicular to v Question What s a vector we know is perpendicular to level curves of g Math 241 Spring 2014 Constrained Optimization and Lagrange Multipliers 14 8 Normal Vectors Idea If we want Dv f x0 y0 0 where v is the tangent vector to the level curve for g we need the gradient f x0 y0 to be perpendicular to v Question What s a vector we know is perpendicular to level curves of g Answer g Math 241 Spring 2014 Constrained Optimization and Lagrange Multipliers 14 8 Lagrange Multipliers Let f g Rn R be functions of n variables To optimize f on a level set for g we look for Critical points Points where f g This works when g 6 0 and if the maximum and minimum exist Math 241 Spring 2014 Constrained Optimization and Lagrange Multipliers 14 8 Example Optimize f x y xy subject to g x y x 2 y 2 1 2 13 2014 https www desmos com calculator Desmos Graphing Calculator 1 3
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