Suppose that the Second Derivatives Test is applied to a real-valued function z = f(x, y), whose second partial derivatives are continuous on a disk D centered at (a, b). af Suppose that (a, b) is a critical point of the functionf such that. (a, b) = 0 and (a, b) = 0. %3D %3D It is found that D(a, b) = -4 and (a, b) = 12 at the critical point (a, b). Which of the following %3D %3D statements is true? O The Second Derivatives test is inconclusive. The function has a local maximum value f(a, b). O The function has a local minimum value f(a, b). O There is a saddle point (a, b, f(a, b)).
Suppose that the Second Derivatives Test is applied to a real-valued function z = f(x, y), whose second partial derivatives are continuous on a disk D centered at (a, b). af Suppose that (a, b) is a critical point of the functionf such that. (a, b) = 0 and (a, b) = 0. %3D %3D It is found that D(a, b) = -4 and (a, b) = 12 at the critical point (a, b). Which of the following %3D %3D statements is true? O The Second Derivatives test is inconclusive. The function has a local maximum value f(a, b). O The function has a local minimum value f(a, b). O There is a saddle point (a, b, f(a, b)).
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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![Suppose that the Second Derivatives Test is applied to a real-valued function \( z = f(x, y) \), whose second partial derivatives are continuous on a disk \( D \) centered at \( (a, b) \).
Suppose that \( (a, b) \) is a critical point of the function \( f \) such that
\[
\frac{\partial f}{\partial x}(a, b) = 0 \quad \text{and} \quad \frac{\partial f}{\partial y}(a, b) = 0.
\]
It is found that \( D(a, b) = -4 \) and
\[
\frac{\partial^2 f}{\partial x^2}(a, b) = 12
\]
at the critical point \( (a, b) \). Which of the following statements is true?
- O The Second Derivatives test is inconclusive.
- O The function has a local maximum value \( f(a, b) \).
- O The function has a local minimum value \( f(a, b) \).
- O There is a saddle point \( (a, b, f(a, b)) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe629c718-b136-4c96-b4c1-19af42ffd554%2F5cbd7186-d154-4ae6-a997-0c56ea86fe6a%2Fr7egc2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that the Second Derivatives Test is applied to a real-valued function \( z = f(x, y) \), whose second partial derivatives are continuous on a disk \( D \) centered at \( (a, b) \).
Suppose that \( (a, b) \) is a critical point of the function \( f \) such that
\[
\frac{\partial f}{\partial x}(a, b) = 0 \quad \text{and} \quad \frac{\partial f}{\partial y}(a, b) = 0.
\]
It is found that \( D(a, b) = -4 \) and
\[
\frac{\partial^2 f}{\partial x^2}(a, b) = 12
\]
at the critical point \( (a, b) \). Which of the following statements is true?
- O The Second Derivatives test is inconclusive.
- O The function has a local maximum value \( f(a, b) \).
- O The function has a local minimum value \( f(a, b) \).
- O There is a saddle point \( (a, b, f(a, b)) \).
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