- Consider The Function F X Y X Y 2 X Y 1 On R A Find All The Critical Points Of F X Y B Notice If 1 (104.41 KiB) Viewed 32 times
Consider the function f(x, y) = x² + y²-2√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 Def is a function of x and y, just like f. So we can take a directional derivative of that directional derivative: let (D/f)(x, y) = D₁(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 = (3), show that 20² f (Df)(x, y) = v²7 +2VzVy ³x əx² 8² f əxəy 20² f Əy² + -2.² for v = (Ur, Uy). (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 r(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 (D(t)f) (F(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}