Problem 1: Recall in question 6 from Exam II I gave you a differentiable functio
ID: 2885815 • Letter: P
Question
Problem 1: Recall in question 6 from Exam II I gave you a differentiable function f(x, y) and set g(u, v) f(u2-2,2u). You then had to use the chain rule to compute gu(u, v) in terms of u and v, and the partial derivatives of f Suppose further that f.(-3, -4) 3 and fy(-3,-4) 2 and that (0,2) is a critical point for f. Answer these additional questions: a) What is the equation (in'u, v) for the tangent line to the level set for g which goes through the point (u, v) (-1,2)? b) In what directions out of (-1,2) is g(u, v) decreasing? c) Find 3 critical points for g. d) Find expressions for gut, gut, g in terms of u, u and the partial derivatives off with respect to x and y (now you might need higher order derivatives). (Hint: in computing gu you get expressions like fz(u2-U2, 2) and you have to find its partial derivatives with respect to u and u. You can use the chain rule again to do this!) e) Suppose (0,2) is a saddle point for f. Must the corresponding critical point (s) for g also be saddle points, regardless of the values of fz (0,2), fry(0, 2) and fy(0,2)?Explanation / Answer
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