Maxima and Minima (§14.7)Math 241, Spring 2014Exam InformationTime and Date Thursday, February 20th, 6:45-8pmLocation Will be sent to you by Tuesday, February 18th.Pencil BRING A NUMBER 2 PENCILConflict Exams Will be scheduled to minimize number ofexams.Website http://www.math.illinois.edu/~jathreya/241/exam1.htmlMath 241, Spring 2014Level sets of the circleMath 241, Spring 2014Level setsClaim∇f is orthogonal to (tangent vectors to) level sets.That is, for a function f : R2→ R, ∇f is perpendicular to tangentlines.For f : R3→ R, ∇f is perpendicular to level surfaces.Math 241, Spring 2014Level setsIdeaSince directional derivatives of f along level sets are 0, wehave, for any tangent vector v to a level set of f ,Dvf = ∇f · v = 0Math 241, Spring 2014Tangent planes to level surfacesWe can use this idea to compute tangent planes to levelsurfaces of f = f (x, y, z):IdeaThe normal vector to the level surface {(x, y, z) : g(x, y, z) = L}at the point (x0, y0, z0) is given by the gradient vector∇g(x0, y0, z0)Math 241, Spring 2014Tangent planes to graphsNote that the graph of a surface z = f (x, y) can be viewed asthe 0−level surface of the functiong(x, y, z) = f (x, y) − z.∇g = (fx, fy, −1)Math 241, Spring 2014Maxima and Minima (§14.7)Local maximaDefinitionf : Rn→ R has an local maximum at the point p if there is aδ > 0 such thatf (p) ≥ f (x) for 0 < kx − pk < δ.Math 241, Spring 2014Maxima and Minima (§14.7)Local 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 2014Maxima and Minima (§14.7)Critical 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 2014Maxima and Minima (§14.7)Critical 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 2014Maxima and Minima (§14.7)Saddle PointsIn higher dimensions, other things can happen:Math 241, Spring 2014Maxima and Minima (§14.7)Potato Chipshttp://freakonomics.com/2011/04/15/the-math-of-pringles/Math 241, Spring 2014Maxima and Minima (§14.7)Second 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 2014Maxima and Minima (§14.7)Second 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 2014Maxima and Minima
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