1. Consider the paraboloid f(x,y) = ax² + bry + cy², where we will assume that a, c# 0. We will investigate the behavior of f at its critical point. (a) Show that (0,0) is the only critical point when b² – 4ac # 0. (b) By completing the square, show that 2 4ас — 6? b x + 2a f(x, y) = 4a2 (c) Suppose that D= 4ac – 6². Without using the second derivative test: i. Suppose that D > 0 and a > 0. Show that f has a local minimum at (0,0). [Hint: Show that f(0,0) = 0. Use the fact that a and D are both positive to conclude that when x,y # 0, f(x, y) > 0.] ii. Suppose that D> 0 and a < 0. Show that ƒ has a local maximum at (0, 0). iii. Finally, suppose that D < 0. Show that f has a saddle point. [Hint: Explain why the tangent plane at (0 0) has equation z = 0 We wish to show that f crosses this tangent plane by showwing there

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