Explanation Given: Thefunction to minimize f ( x , y ) = x 2 − y 2 subject to x 2 + y 2 = 4 Concept Used: To optimize (i.e., to find the minimum and maximum value of) a function f ( x , y ) subject to the constraint g ( x , y ) = k Solve the following system of equations ∇ f ( x , y ) = λ ∇ g ( x , y ) g ( x , y ) = k where λ is the Lagrange Multiplier. Plug all the solutions ( x , y ) from the above step into f ( x , y ) . The maximum value of the function f ( x , y ) subject to the constraint g ( x , y ) = k is the maximum value of f ( x , y ) after plugging all the solutions ( x , y ) into f ( x , y ) . Calculation: For the givenfunction f ( x , y ) = x 2 − y 2 Calculating the first partial derivatives f x and f y of f ( x , y ) f ( x , y ) = x 2 − y 2 i.e., f x ( x , y ) = 2 x and, f y ( x , y ) = − 2 y and, calculating the first partial derivatives g x and g y of g ( x , y ) g ( x , y ) = x 2 + y 2 i.e., g x ( x , y ) = 2 x and, g y ( x , y ) = 2 y Substituting the values in the equation f x ( x , y ) = λ g x ( x , y ) i.e., 2 x = λ ( 2 x ) 2 x − λ ( 2 x ) = 0 ( 2 x ) ( 1 − λ ) = 0 i

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

See similar textbooks

Videos

Question

Chapter 11.8, Problem 11PS

To determine

Tominimize f(x,y)=x2−y2 subject to constraint x2+y2=4 using the method of Lagrange multipliers.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.8, Problem 10PS

Chapter 11.8, Problem 12PS

Students have asked these similar questions

5:38 Video Message instructor Submit Question ||| D

8:38 *** TEMU TEMU -3 -2 7 B 2 1 & 5G. 61% 1 2 -1 Based on the graph above, determine the amplitude, period, midline, and equation of the function. Use f(x) as the output. Amplitude: 2 Period: 2 Midline: 2 ☑ syntax error: this is not an equation. Function: f(x) = −2 cos(πx + 2.5π) +2× Question Help: Worked Example 1 ☑ Message instructor Submit Question ||| <

8:39 *** TEMU 5G 60% A ferris wheel is 28 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. What is the amplitude? 14 meters What is the equation of the Midline? y = 16 What is the period? 4 meters minutes The equation that models the height of the ferris wheel after t minutes is: f(t): = ƒ (3) = ·−14(0) + 16 syntax error: you gave an equation, not an expression. syntax error. Check your variables - you might be using an incorrect one. How high are you off of the ground after 3 minutes? Round your answe the nearest meter. ||| <

Chapter 11 Solutions

Calculus

Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Tominimize f ( x , y ) = x 2 − y 2 subject to constraint x 2 + y 2 = 4 using the method of Lagrange multipliers.

Solution Summary: The author explains how to minimize f(x,y) using the method of Lagrange multipliers.

Videos

Tominimize f(x,y)=x2−y2 subject to constraint x2+y2=4 using the method of Lagrange multipliers.

Want to see the full answer?

Chapter 11 Solutions