Problem 5 - a) If all second order partial derivatives of f(x, y, z) are continuous in R³, Are the following statements true? Why? Vf(0,0,0) = (0,0,0), and fyy(0,0, 0) < 0, then (0,0,0) is not a point of local minimum of this function.

Problem 5 - a) If all second order partial derivatives of f(x, y, z) are continuous in R³, Are the following statements true? Why? Vf(0,0,0) = (0,0,0), and fyy(0,0, 0) < 0, then (0,0,0) is not a point of local minimum of this function.

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

Are the following statements true? Why?
a) If all second order partial derivatives of f(x, y, z) are continuous in R³,
Vf(0,0,0) = (0,0,0), and fyy (0,0, 0) < 0,
Problem 5
then (0,0,0) is not a point of local minimum of this function.
b) A point (1,0,0) is a saddle point of the function f(x, y, z) = (x – 1)²yz.
c) The quadratic form (V²f(x, y, z)h, h) for the function
f (x, y, z) = Ax² + y² + z² + 2y + 2z is positive - definite if A > 0.

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