Question

Please do question 5a and 5b

4. In this problem we analyze the behavior of the polynomial f (x, y) = ax² + bxy + cy? (without using the Second Derivatives Test) by identifying the graph as a paraboloid. (a) By completing the square, show that if a + 0, then b 2 4ac - 62 f(x, y) = ax² + bxy + cy? = a [( 2 + Y + 2a 4a2 (b) Let D = 4ac – 62. Show that if D > 0 and a > 0, then f has a local minimum at (0,0). (C) Show that if D > 0 and a < 0, then f has a local maximum at (0,0). (d) Show that if D< 0, then (0,0) is a saddle point. 5. (a) Suppose f is any function with continuous second-order partial derivatives such that f(0,0) = 0 and (0,0) is a critical point of f. Write an expression for the second-degree Taylor polynomial, Q, of f at (0,0). (b) What can you conclude about Q from Problem 4?

Answer 1

5. @o) is a critical point of f. fe 00)= 0 - fy Coo), Thus, the second degree taylor's polynomial Q about (60) is give or by f(20) +x. fz(20) + fy (0,0) + 2 [ a fax (0,0) + zay fyn Coo) + 9 fyg (oo) + [A²+ Bay tegh [where Az fece (0) B = 2 fyn C0,0), c= fyl9o] 2

can easily Q. using problem (4), now we et conclude about 27 A([x+ Fy] 7 1ACB" y] (we're assumed that Axo) Now if 6 AXO, 4AC-B>o. then Qyo for attains all (ny) (0,0). So, in this case its at (6,o) if A TO, 4 AC-B²>0; then Q<o for all (2,4) (@o). So, in this case Q altains its local maxima at 699 (111) And if if D <o, D<o, then a saddle point at (0,0) D = 4 AC- minima it has B² where