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(B) Let f be a function of x and y with fx (x, yo) = 0 and fy (x,yo) = 0 for some (xo. Yo) € dom f. Suppose that the second-order partial derivatives of f are continuous on a disk centered at (x, yo). Let D(xo, Yo) = fxx (xo, Yo) fyy(xo. Yo) - [fry (xo, Yo)]². 1. If D (a, b) > 0 and fxx (a, b) > 0 then (a, b) is a local minimum of f. 2. If D (a, b) > 0 and fxx(a, b) < 0 then (a, b) is a local maximum of f.

(B) Let f be a function of x and y with fx (x, yo) = 0 and fy (x,yo) = 0 for some (xo. Yo) € dom f. Suppose that the second-order partial derivatives of f are continuous on a disk centered at (x, yo). Let D(xo, Yo) = fxx (xo, Yo) fyy(xo. Yo) - [fry (xo, Yo)]². 1. If D (a, b) > 0 and fxx (a, b) > 0 then (a, b) is a local minimum of f. 2. If D (a, b) > 0 and fxx(a, b) < 0 then (a, b) is a local maximum of f.

Advanced Engineering Mathematics
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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(B) Let f be a function of x and y with fx (xo, Yo) = 0 and f(x,y₁) = 0 for some (xo. Yo) € dom f. Suppose that the second-order partial derivatives of f are continuous on a disk centered at (xo, Yo). Let D(xo. Yo) = fxx (xo, Yo) fyy(xo, Yo) - [fry (xo, Yo)]². 1. If D(a, b) > 0 and fxx(a, b) > 0 then (a, b) is a local minimum of f. 2. If D (a, b) > 0 and fxx(a, b) < 0 then (a, b) is a local maximum of f. 3. If D (a, b) < 0 then (a, b) is a saddle point of f. 4. If D (a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (the test is inconclusive) [1]. The goal of this study is to show alternative condition/s when the second partial derivative test asserts D(a, b) = 0 or an inconclusive result. This will lead to a result of either a local minimum of for local maximum of for saddle point of f.
Transcribed Image Text:(B) Let f be a function of x and y with fx (xo, Yo) = 0 and f(x,y₁) = 0 for some (xo. Yo) € dom f. Suppose that the second-order partial derivatives of f are continuous on a disk centered at (xo, Yo). Let D(xo. Yo) = fxx (xo, Yo) fyy(xo, Yo) - [fry (xo, Yo)]². 1. If D(a, b) > 0 and fxx(a, b) > 0 then (a, b) is a local minimum of f. 2. If D (a, b) > 0 and fxx(a, b) < 0 then (a, b) is a local maximum of f. 3. If D (a, b) < 0 then (a, b) is a saddle point of f. 4. If D (a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (the test is inconclusive) [1]. The goal of this study is to show alternative condition/s when the second partial derivative test asserts D(a, b) = 0 or an inconclusive result. This will lead to a result of either a local minimum of for local maximum of for saddle point of f.
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