Consider a function f (x, y) whose domain is R' and whose first-order partial derivatives both exist. • If f (x, y) has a local maximum at (a, b). then f. (a, b) and fy(a, b)= If f (z, y) has a local minimum at (a, b). then f. (a, b) and f,(a, b) If f (r, y) has a saddle point at (a, b), then f. (a, b) and fy(a, b)= Suppose that f(r, y) has a critical point at (a, b) and that all second-order partial derivatives of f(r, y) exist and are continuous at (a, b). (Se (a,6) Teyd,0)) = S. (a, b) fw (a, b) –Say (a, b)". Syn (a, b) fy (a, b), Let D= det = f. (a, b) fw (a, b)-Vay (a, b)". • {(z, y) necessarily has a local minimum at (a, b) it D ISelect ) and f(a, b) (Select] and • f(r, y) necessarily has a local maximum at (a, b) if D ISelect) fa (a, b) ISelect] f(x, y) necessarily has a saddle point at (a, b) if D ( Select]
Consider a function f (x, y) whose domain is R' and whose first-order partial derivatives both exist. • If f (x, y) has a local maximum at (a, b). then f. (a, b) and fy(a, b)= If f (z, y) has a local minimum at (a, b). then f. (a, b) and f,(a, b) If f (r, y) has a saddle point at (a, b), then f. (a, b) and fy(a, b)= Suppose that f(r, y) has a critical point at (a, b) and that all second-order partial derivatives of f(r, y) exist and are continuous at (a, b). (Se (a,6) Teyd,0)) = S. (a, b) fw (a, b) –Say (a, b)". Syn (a, b) fy (a, b), Let D= det = f. (a, b) fw (a, b)-Vay (a, b)". • {(z, y) necessarily has a local minimum at (a, b) it D ISelect ) and f(a, b) (Select] and • f(r, y) necessarily has a local maximum at (a, b) if D ISelect) fa (a, b) ISelect] f(x, y) necessarily has a saddle point at (a, b) if D ( Select]
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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can you plz slove it ASAP, I'll like it
![Consider a function f (2, y) whose domain is R? and whose first-order partial derivatives both exist.
• If f (r, y) has a local maximum at (a, b), then f-(a, b)
and fy(a, b) =
• If f (z, y) has a local minimum at (a, b), then f- (a, b) =
and fy(a, b) =
• If f (r, y) has a saddle point at (a, b), then f. (a, b) =
and fy(a, b) =
Suppose that f(x, y) has a critical point at (a, b) and that all second-order partial derivatives of
f(r, y) exist and are continuous at (a, b).
frz (a, b) fry (a, b)
fyz (a, b) fy(a, b))
Let D = det
= fz (a, b) fyy (a, b) – [Szy (a, b)]².
• f(z, y) necessarily has a local minimum at (a, b) if D I Select)
* and fz (a, b)
[ Select ]
and
f(r, y) necessarily has a local maximum at (a, b) if D I Select]
fz (a, b) I Select)
• f(z, y) necessarily has a saddle point at (a, b) if D [Select]
Suppose f(z, y, 2) is continuous on the set
B = {(z, y, 2)| –1552,0<y 900,-10 < z S-9.9}.
Then f(z, y, z) necessarily attains an absolute maximum value somewhere on B:
ISelect |
f(z, y, 2) necessarily attains an absolute minimum value somewhere on B:
( Select)
f(z, y, 2) necessarily has a saddle point somewhere on B: ISelect]
Suppose g(r, y, z) is continuous on all of R". Then g(x, y, z) nekessarily attains an absolute
maximum value somewhere in R: I Select )
Suppose h(z, y, z) is continuous on S = {(x,y, z)|a +y + < 4}. Then h(r, y, 2) necessarily
attains an absolute maximum value somewhere in S: I Select ]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21da0129-2278-4663-a448-2a7163c389d5%2F9f484eed-4aeb-47ff-ac65-69aadc0a71f8%2F96l28rh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a function f (2, y) whose domain is R? and whose first-order partial derivatives both exist.
• If f (r, y) has a local maximum at (a, b), then f-(a, b)
and fy(a, b) =
• If f (z, y) has a local minimum at (a, b), then f- (a, b) =
and fy(a, b) =
• If f (r, y) has a saddle point at (a, b), then f. (a, b) =
and fy(a, b) =
Suppose that f(x, y) has a critical point at (a, b) and that all second-order partial derivatives of
f(r, y) exist and are continuous at (a, b).
frz (a, b) fry (a, b)
fyz (a, b) fy(a, b))
Let D = det
= fz (a, b) fyy (a, b) – [Szy (a, b)]².
• f(z, y) necessarily has a local minimum at (a, b) if D I Select)
* and fz (a, b)
[ Select ]
and
f(r, y) necessarily has a local maximum at (a, b) if D I Select]
fz (a, b) I Select)
• f(z, y) necessarily has a saddle point at (a, b) if D [Select]
Suppose f(z, y, 2) is continuous on the set
B = {(z, y, 2)| –1552,0<y 900,-10 < z S-9.9}.
Then f(z, y, z) necessarily attains an absolute maximum value somewhere on B:
ISelect |
f(z, y, 2) necessarily attains an absolute minimum value somewhere on B:
( Select)
f(z, y, 2) necessarily has a saddle point somewhere on B: ISelect]
Suppose g(r, y, z) is continuous on all of R". Then g(x, y, z) nekessarily attains an absolute
maximum value somewhere in R: I Select )
Suppose h(z, y, z) is continuous on S = {(x,y, z)|a +y + < 4}. Then h(r, y, 2) necessarily
attains an absolute maximum value somewhere in S: I Select ]
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