Explanation Given Information: The function is f ( x , y ) = x 4 + y 4 + 4 x − 32 y + 80 . Formula used: For a function, f ( x ) the partial derivative with respect to x is calculated as, the chain rule states that, ∂ ∂ x ( f ( x ) ) n = n ( f ( x ) ) n − 1 ∂ ∂ x [ f ( x ) ] Here, f ( x ) is the function of x and n is the real number. Condition of relative maximum and minimum: D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 If the function f has two variables. It has a relative minimum at ( a , b ) if D > 0 and f x x ( a , b ) > 0 . If the function f has two variables. It has a relative maximum at ( a , b ) if D > 0 and f x x ( a , b ) < 0 . Calculation: Partially differentiate the function f ( x , y ) = 3 x − 6 y − x 2 − y 2 for the values f x , f y , f x x , f y y and f x y . Consider the function, f ( x , y ) = 3 x − 6 y − x 2 − y 2 …… (1) Partially differentiate the function for x , f x = ∂ ∂ x ( x 4 + y 4 + 4 x − 32 y + 80 ) = ∂ ∂ x ( x 4 ) + ∂ ∂ x ( y 4 ) + ∂ ∂ x ( 4 x ) − ∂ ∂ x ( 32 y ) + ∂ ∂ x ( 80 ) = 4 x 3 − 0 + 4 − 0 + 0 = 4 x 3 + 4 …… (2) Partially differentiate the function (1) for y . f y = ∂ ∂ y ( x 4 + y 4 + 4 x − 32 y + 80 ) = ∂ ∂ y ( x 4 ) + ∂ ∂ y ( y 4 ) + ∂ ∂ y ( 4 x ) − ∂ ∂ y ( 32 y ) + ∂ ∂ y ( 80 ) = 0 + 4 y 3 + 0 − 32 + 0 = 4 y 3 − 32 …… (3) Perform partial differentiation with respect to x on the function (2). f x x = ∂ ∂ x ( 4 x 3 + 4 ) = ∂ ∂ x ( 4 x 3 ) + ∂ ∂ x ( 4 ) = 12 x 2 + 0 = 12 x 2 Perform partial differentiation with respect to y on the function (3)

Calculus and Its Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Bittinger, Ellenbogen & Surgent, The Calculus and Its Applications Series)

11th Edition

ISBN: 9780133795561

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Chapter 6, Problem 26RE

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To calculate: The relative maximum and minimum values for the function f(x,y)=x4+y4+4x−32y+80.

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Chapter 6, Problem 25RE

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Calculus and Its Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Bittinger, Ellenbogen & Surgent, The Calculus and Its Applications Series)

Ch. 6.1 - Forf(x,y)=x23xy,find(0,2),f(2,3),andf(10,5).Ch. 6.1 - 2. . Ch. 6.1 - 3. . Ch. 6.1 - 4. . Ch. 6.1 - Forf(x,y)=Inx+y3,findf(e,2),f(e2,4),andf(e3,5).Ch. 6.1 - 6. . Ch. 6.1 - Forf(x,y,z)=x2y2+z2,findf(1,2,3)andf(2,1,3).Ch. 6.1 - 8. . Ch. 6.1 - In Exercises 9-14, determine the domain of each...Ch. 6.1 - In Exercises 9-14, determine the domain of each...

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The relative maximum and minimum values for the function f ( x , y ) = x 4 + y 4 + 4 x − 32 y + 80 .

Solution Summary: The author calculates the relative maximum and minimum values for the function f(x,y).