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23-26 True-False Determine whether the statement is true or false. Explain your answer. In these exercises, assume that f(x. y) has continuous second-order partial derivatives and that D(x, y) fax(x. y) fy (x. y)- (x. y) 23. If the function f is defined on the disk x + y? < 1, then f has a critical point somewhere on this disk. %3D 24. If the function f is defined on the disk x2 + y2 s 1, and if f is not a constant function, then f has a finite number of critical points on this disk.

23-26 True-False Determine whether the statement is true or false. Explain your answer. In these exercises, assume that f(x. y) has continuous second-order partial derivatives and that D(x, y) fax(x. y) fy (x. y)- (x. y) 23. If the function f is defined on the disk x + y? < 1, then f has a critical point somewhere on this disk. %3D 24. If the function f is defined on the disk x2 + y2 s 1, and if f is not a constant function, then f has a finite number of critical points on this disk.

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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23-26 True-False Determine whether the statement is true or false. Explain your answer. In these exercises, assume that f(x. y) has continuous second-order partial derivatives and that D(x, y) fax(x. y) fy (x, y)- (r. y) 23. If the function f is defined on the disk x + y? < 1, then f has a critical point somewhere on this disk. 24. If the function f is defined on the disk x²+ y2 < 1, and if f is not a constant function, then f has a finite number of critical points on this disk. 25. If P(xo. yo) is a critical point of f, and if f is defined on a disk centered at P with D(xo. yo) > 0, then f has a relative extremum at P. 47
Transcribed Image Text:23-26 True-False Determine whether the statement is true or false. Explain your answer. In these exercises, assume that f(x. y) has continuous second-order partial derivatives and that D(x, y) fax(x. y) fy (x, y)- (r. y) 23. If the function f is defined on the disk x + y? < 1, then f has a critical point somewhere on this disk. 24. If the function f is defined on the disk x²+ y2 < 1, and if f is not a constant function, then f has a finite number of critical points on this disk. 25. If P(xo. yo) is a critical point of f, and if f is defined on a disk centered at P with D(xo. yo) > 0, then f has a relative extremum at P. 47
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