1. Nonisolated Extrema (20 points) Consider the function f(x, y) = x² + y² - 2₁ x² + y² +1 on R². (a) Find all the criti

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1 Nonisolated Extrema 20 Points Consider The Function F X Y X Y 2 X Y 1 On R A Find All The Criti 1 (526.57 KiB) Viewed 81 times

1. Nonisolated Extrema (20 points) Consider the function f(x, y) = x² + y² - 2₁ x² + y² +1 on R². (a) Find all the critical points of f(x, y). (b) Notice (if done correctly) the strange situation that, for this particular function, most of the critical points form a curve. Use the Chain Rule to show that this curve must be a level curve of f. (c) Notice that for a fixed vector the directional derivative Düf is a function of x and y, just like f. So we can take a directional derivative of that directional derivative: let (D¾ƒ)(x, y) := Dû(D=f(x,y)) denote the directional second derivative of f in the direction 7. Recalling that directional derivatives can be computed by the rule D f = Vf and for now just writing Vf = af af მ· მყ show that = (Vx, vy). for v = 20² f (D²/f)(x, y) = √²/√x² 8² f + 2vхvy Әхду a² f + v² ay ² (d) Compute all second partial derivatives of f(x, y). (e) Find a parametrization r(t) of the level curve of critical points from the first part and evaluate the second partial derivatives at the components x(t), y(t). (f) Classify all local extrema of f in D: 1. Note that the second derivative test will not work at any of these critical points instead use the previous three parts to compute (D²/(t)f)(r(t)) along the critical point curve and argue by the sign of that function). 2. Argue that, being surrounded by local minima and no other critical points, the re- maining critical point must be a local maximum. (g) Determine all extrema of f on the filled-in ellipse D = {(x, y) : x² + 2y² ≤ 1}