(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). [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 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.]
(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). [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 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.]
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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Can I get c and d answered?
![1. Consider the paraboloid f(x, y)
behavior of f at its critical point.
ax? + bry + cy², where we will assume that a, c+ 0. We will investigate the
(a) Show that (0,0) is the only critical point when 62
- 4ac + 0.
(b) By completing the square, show that
2
4ас — b2
f(x, y) :
= a
x +
2a
4а2
(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).
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 ƒ 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1863dd2d-66ae-4de0-8154-3d7d6de9d0c9%2Ffb6d2fd4-6ac0-4600-a7bc-002f52564c97%2Fptvp8ph_processed.png&w=3840&q=75)
Transcribed Image Text:1. Consider the paraboloid f(x, y)
behavior of f at its critical point.
ax? + bry + cy², where we will assume that a, c+ 0. We will investigate the
(a) Show that (0,0) is the only critical point when 62
- 4ac + 0.
(b) By completing the square, show that
2
4ас — b2
f(x, y) :
= a
x +
2a
4а2
(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).
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 ƒ 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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