Explanation Critical points of the function are referred to points from its domain for which the derivative of the function is 0. Mathematically, we can say that d f d x ( x , y ) = 0 and d f d y ( x , y ) = 0 Given: A function f ( x , y ) = x 4 − 2 x 2 + y 2 − 6 y Formula Used: Second Derivative Test: Determinant of a function is given as D ( x , y ) = f x x ( x , y ) f y y ( x , y ) − ( f x y ( x , y ) ) 2 1. Maxima Point: If D ( a , b ) > 0 and f x x ( a , b ) > 0 , then ( a , b ) is a local maxima point 2. Minima Point: If D ( a , b ) > 0 and f x x f x x ( a , b ) < 0 , then ( a , b ) is a local minima point 3. Maxima Point: If D ( a , b ) < 0 then ( a , b ) is a Saddle point 4. If D ( a , b ) = 0 , the second derivative test is not decisive, and the point ( a , b ) can be any of a saddle, maxima or minima point. Calculation: The given function is f ( x , y ) = x 4 − 2 x 2 + y 2 − 6 y Derivative of the function is given as d y d x ( x , y ) = ( d f d x , d f d y ) f x = d d x ( x 4 − 2 x 2 + y 2 − 6 y ) = 4 x 3 − 4 x + ( 0 ) − 6 ( 0 ) = 4 x 3 − 4 x f y = d d y ( x 4 − 2 x 2 + y 2 − 6 y ) = (0) − 2 ( 0 ) + 2 y − 6 = 2 y − 6 Equating f x and f y equal to zero to get the critical points... To determine (b) Find the minimum value of the function f ( x , y ) = x 4 − 2 x 2 + y 2 − 6 y by completing the square

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Question

Chapter 14, Problem 46CRE

To determine

(a)

Find the critical points of the function $f (x, y) = x^{4} - 2 x^{2} + y^{2} - 6 y$ and analyze the critical points using the Second Derivative Test.

To determine

(b)

Find the minimum value of the function $f (x, y) = x^{4} - 2 x^{2} + y^{2} - 6 y$ by completing the square

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Chapter 14, Problem 45CRE

Chapter 14, Problem 47CRE

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Chapter 14 Solutions

Calculus: Early Transcendentals

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

Knowledge Booster

(a) Find the critical points of the function f ( x , y ) = x 4 − 2 x 2 + y 2 − 6 y and analyze the critical points using the Second Derivative Test.

(a) Find the critical points of the function f(x,y)=x4−2x2+y2−6y and analyze the critical points using the Second Derivative Test.

(b) Find the minimum value of the function f(x,y)=x4−2x2+y2−6y by completing the square

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Chapter 14 Solutions

(a)

Find the critical points of the function $f (x, y) = x^{4} - 2 x^{2} + y^{2} - 6 y$ and analyze the critical points using the Second Derivative Test.

(b)

Find the minimum value of the function $f (x, y) = x^{4} - 2 x^{2} + y^{2} - 6 y$ by completing the square