Homework Help is Here – Start Your Trial Now!
Math
Calculus
ple Find the local extreme values of the function (x, y) = xy - x - y? - 2r - 2y + 4.

ple Find the local extreme values of the function (x, y) = xy - x - y? - 2r - 2y + 4.

Calculus: Early Transcendentals
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...
See similar textbooks
icon
Related questions
Question
Example
(NITC) Local maxima and local minima of functions of EXAMPLE • Example Find the local extreme values of f(x, y) = x² + y? – 4y + 9. Solution The domain of f is the entire plane (so there are no boundary points) and the partial derivatives f = 2x and fy = 2y - 4 exist everywhere. Therefore, local extreme values can occur only where f = 2x = 0 and , = 2y - 4 = 0. The only possibility is the point (0, 2), where the value of f is 5. Since f(x, v) = x* + (v - 2) + 5 is never less than 5, we see that the critical point (0, 2) gives a local minimum 15 The graph of the function J(x, y) = x + y* - 4y + 9 is a paraboloid which has a local minimum value of 5 at the point (0, 2) (NITC) Local maxima and local minima of functions of 6/9 THE SECOND PARTIALS TEST 13.8.6 THEOREM (The Second Partials Test) Let f be a function of two variables with continuous second-order partial derivatives in some disk centered at a critical point (Xo. yo), and let D = fr«(xo. yo) fyy(xo. yo) - f(xo. yo) (a) If D > 0 and f(xo. yo) > 0, then f has a relative minimum at (xp. yo). (b) If D > 0 and fa(xo. yo) < 0, then f has a relative maximum at (xo, yo). (c) If D < 0, then f has a saddle point at (xp. Vo). (d) If D = 0, then no conclusion can be drawn. The expression fnfw - fn is called the discriminant or Hessian of f. It is some- times easier to remember it in determinant form, fsfy - fw = iv I Say fyl (NITC) Local maxima and local minima of functions of 7/9 EXAMPLE > Example Find the local extreme values of the function f(x, y) = xy - x? - v2 - 2x - 2y + 4. >> (NITC) Local maxima and local minima of functions of 8/9 LXAMPLE • Example Locate all relative extrema and saddle points of f(x, y) = 4xy –-x- y 1-10 y = * = y f.(x, y) = 4y – 4x = 0 1-20 1-30 f,(x, y) = 4x - 4y³ = 0 1-40 yields x = (x')* or x(x – 1) = 0, which has solutions x = 0, x = 1, x = -I. the corresponding y-values y = 0, y = 1, y = -1. the critical points of f are (0, 0), (1, 1), and (-1. -1).
Transcribed Image Text:(NITC) Local maxima and local minima of functions of EXAMPLE • Example Find the local extreme values of f(x, y) = x² + y? – 4y + 9. Solution The domain of f is the entire plane (so there are no boundary points) and the partial derivatives f = 2x and fy = 2y - 4 exist everywhere. Therefore, local extreme values can occur only where f = 2x = 0 and , = 2y - 4 = 0. The only possibility is the point (0, 2), where the value of f is 5. Since f(x, v) = x* + (v - 2) + 5 is never less than 5, we see that the critical point (0, 2) gives a local minimum 15 The graph of the function J(x, y) = x + y* - 4y + 9 is a paraboloid which has a local minimum value of 5 at the point (0, 2) (NITC) Local maxima and local minima of functions of 6/9 THE SECOND PARTIALS TEST 13.8.6 THEOREM (The Second Partials Test) Let f be a function of two variables with continuous second-order partial derivatives in some disk centered at a critical point (Xo. yo), and let D = fr«(xo. yo) fyy(xo. yo) - f(xo. yo) (a) If D > 0 and f(xo. yo) > 0, then f has a relative minimum at (xp. yo). (b) If D > 0 and fa(xo. yo) < 0, then f has a relative maximum at (xo, yo). (c) If D < 0, then f has a saddle point at (xp. Vo). (d) If D = 0, then no conclusion can be drawn. The expression fnfw - fn is called the discriminant or Hessian of f. It is some- times easier to remember it in determinant form, fsfy - fw = iv I Say fyl (NITC) Local maxima and local minima of functions of 7/9 EXAMPLE > Example Find the local extreme values of the function f(x, y) = xy - x? - v2 - 2x - 2y + 4. >> (NITC) Local maxima and local minima of functions of 8/9 LXAMPLE • Example Locate all relative extrema and saddle points of f(x, y) = 4xy –-x- y 1-10 y = * = y f.(x, y) = 4y – 4x = 0 1-20 1-30 f,(x, y) = 4x - 4y³ = 0 1-40 yields x = (x')* or x(x – 1) = 0, which has solutions x = 0, x = 1, x = -I. the corresponding y-values y = 0, y = 1, y = -1. the critical points of f are (0, 0), (1, 1), and (-1. -1).
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS