Absolute ExtremaConstrained Optimization and Lagrange Multipliers (§14.8)Math 241, Spring 2014Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Math 241, Spring 2014Local minimaDefinitionf : Rn→ R has an local minimum at the point p if there is aδ > 0 such thatf (p) ≤ f (x) for 0 < kx − pk < δ.Math 241, Spring 2014Critical PointsIdeaIf any component of the gradient vector is non-zero, thefunction will change (increase/decrease) in that directionSo, local maxima and minima of a function f happen at placeswhere ∇f = 0.Math 241, Spring 2014Critical PointsDefinitionThe critical points of the function f are exactly the places where∇f = 0QuestionHow can we tell when a critical point is a minimum or amaximum?Math 241, Spring 2014Saddle PointsIn higher dimensions, other things can happen:Math 241, Spring 2014Potato Chipshttp://freakonomics.com/2011/04/15/the-math-of-pringles/Math 241, Spring 2014Second derivative test, 2-variablesLike in 1-variable, we have an second derivative test. Forf = 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))2Math 241, Spring 2014Second derivative test, 2-variablesTheoremLet (a, b) be a critical point for the function f : R2→ R, i.e.,∇f (a, b) = 0. IfD > 0, fxx(a, b) > 0 f (a, b) is a local minimum for fD > 0, fxx(a, b) < 0 f (a, b) is a local maximum for fD < 0 f (a, b) is neither a local maximum or minimum for fMath 241, Spring 2014ExamplesMath 241, Spring 2014Absolute ExtremaClosed and Bounded SetsDefinitionA set A ⊂ Rnis closed if it contains all of its boundary points.DefinitionA set A ⊂ Rnis bounded if it can be contained in a ball of finiteradius.Math 241, Spring 2014Absolute ExtremaExtremaTheoremIf f is a continuous function, it attains an absolute maximumand an absolute minimum on the set A.Math 241, Spring 2014Absolute ExtremaLooking for extremaLook at critical points, look at boundary.QuestionHow to deal with the boundary?Math 241, Spring 2014Constrained Optimization and Lagrange Multipliers (§14.8)Constrained OptimizationIdeaOften, we’re interested in optimizing one function f (x, y) on aregion defined by another function g(x, y). That is, we want tofind extrema of f on the regionA = {(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 2014Constrained Optimization and Lagrange Multipliers (§14.8)Constrained OptimizationQuestionHow do we optimize the function f (x, y) along the level curve{(x, y) : g(x, y) = L}?Math 241, Spring 2014Constrained Optimization and Lagrange Multipliers (§14.8)GradientsIdeaWe 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 thetangent line to the curve. That is, we want to find (x0, y0) wherethe tangent vector v to the level curve hasDvf (x0, y0) = 0Math 241, Spring 2014Constrained Optimization and Lagrange Multipliers (§14.8)Normal VectorsIdeaIf we want Dvf (x0, y0) = 0, where v is the tangent vector to thelevel curve for g, we need the gradient ∇f (x0, y0) to beperpendicular to v.QuestionWhat’s a vector we know is perpendicular to level curves of g?Answer: ∇g.Math 241, Spring 2014Constrained Optimization and Lagrange Multipliers (§14.8)Normal VectorsIdeaIf we want Dvf (x0, y0) = 0, where v is the tangent vector to thelevel curve for g, we need the gradient ∇f (x0, y0) to beperpendicular to v.QuestionWhat’s a vector we know is perpendicular to level curves of g?Answer: ∇g.Math 241, Spring 2014Constrained Optimization and Lagrange Multipliers (§14.8)Lagrange MultipliersLet f , g : Rn→ R be functions of n variables. To optimize f on alevel set for g, we look for:Critical points Points where ∇f = λ∇g.This works when ∇g 6= 0, and if the maximum and minimumexist.Math 241, Spring 2014Constrained Optimization and Lagrange Multipliers (§14.8)ExampleOptimize f (x, y) = xy subject to g(x, y) = x2+ y2= 12/13/2014 Desmos Gra phing Calculatorhttps://ww w.desmos.com/calcula tor
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