1. Consider the paraboloid f(x,y) = ax² + bxy + cy2, where we will assume that a, c 0. We will investigate thebehavior of f at its critical point.(a) Show that (0,0) is the only critical point when b24ас + 0.(b) By completing the square, show that24ас62f (x, y)= a2a4а2(c) Suppose that D=4ac – b2. Without using the second derivative test:i. Suppose that D > 0 and a > 0. Show that f has a local minimum at (0,0).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 f 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 planeat (0,0) has equation z = 0. We wish to show that f crosses this tangent plane, by showing thereexist different paths for which f has opposite signs along those paths.][Hint: Show that(d) Point out that the value D as described above is exactly the function D(x, y) involved in the secondderivative test, and that the results match that test.

As per the guidelines of Bartleby for more than one question asked, only two is to be answered, you…

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

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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

1. Consider the paraboloid f(x,y) = ax² + bxy + 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
4ас + 0.
(b) By completing the square, show that
2
4ас
62
f (x, y)
= a
2a
4а2
(c) Suppose that D=
4ac – b2. Without using the second derivative test:
i. Suppose that D > 0 and a > 0. Show that f has a local minimum at (0,0).
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 f 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 showing there
exist different paths for which f has opposite signs along those paths.]
[Hint: Show that
(d) Point out that the value D as described above is exactly the function D(x, y) involved in the second
derivative test, and that the results match that test.

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