Explanation Given information: The function is f ( x , y ) = x 3 + y 2 − 3 x − 8 y + 12 . Explanation: Use the First-Derivative test for functions of two variables: If f ( x , y ) has either a relative maximum or minimum at point ( x , y ) = ( a , b ) , then ∂ f ∂ x ( a , b ) = 0 and ∂ f ∂ y ( a , b ) = 0 . Find the first partial derivatives ∂ f ∂ x and ∂ f ∂ y of the f ( x , y ) = x 3 + y 2 − 3 x − 8 y + 12 . Differentiate f ( x , y ) = x 3 + y 2 − 3 x − 8 y + 12 with respect to x by treating y as constant. ∂ f ∂ x = ∂ ∂ x ( x 3 + y 2 − 3 x − 8 y + 12 ) ⇒ ∂ f ∂ x = 3 x 2 − 3 Differentiate f ( x , y ) = x 3 + y 2 − 3 x − 8 y + 12 with respect to y by treating x as constant. ∂ f ∂ y = ∂ ∂ y ( x 3 + y 2 − 3 x − 8 y + 12 ) ⇒ ∂ f ∂ y = 2 y − 8 By using the First-Derivative test, set the first partial derivatives equal to zero and solve the system of equations. ∂ f ∂ x = 0 3 2 x − 3 = 0 ⇒ 3 x 2 = 3 ⇒ x 2 = 1 ⇒ x = ± 1 Similarly, ∂ f ∂ y = 0 . 2 y − 8 = 0 ⇒ 2 y − 8 = 0 ⇒ 2 y = 8 ⇒ y = 4 Therefore, the only possible relative maximum or minimum point of the f ( x , y ) = x 3 + y 2 − 3 x − 8 y + 12 are ( 1 , 4 ) and ( − 1 , 4 ) Use the second-derivative test to determine the nature of the f ( x , y ) = x 3 + y 2 − 3 x − 8 y + 12 at the point ( 1 , 4 ) and ( − 1 , 4 ) . The Second-Derivative test: Suppose that f ( x , y ) is a function and ( a , b ) is a point at which ∂ f ∂ x ( a , b ) = 0 and ∂ f ∂ y ( a , b ) = 0 and D ( x , y ) = ∂ 2 f ∂ x 2 × ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2 . There are four conditions to determine a relative maximum and minimum as follows: 1. If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 , then f ( x , y ) has a relative minimum at ( a , b ) . 2. If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 , then f ( x , y ) has a relative maximum at ( a , b ) . 3. If D ( a , b ) < 0 , then f ( x , y ) has neither a relative maximum nor a relative minimum at ( a , b )

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Chapter 7, Problem 23RE

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All the possible relative maximum or minimum points (x, y) of the f(x,y)=x3+y2−3x−8y+12, then use the second-derivative test to determine the nature of f(x,y) at each of these points. If the second-derivative test is inconclusive, then state it.

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Chapter 7, Problem 22RE

Chapter 7, Problem 24RE

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Chapter 7 Solutions

Calculus & Its Applications (14th Edition)

Ch. 7.1 - Let f(x,y,z)=x2+y(xz)4. Compute f(3,5,2).Ch. 7.1 - Prob. 2CYU Ch. 7.1 - Let f(x,y)=x23xyy2. Compute f(5,0),f(5,2) and...Ch. 7.1 - Prob. 2E Ch. 7.1 - Let g(x,y,z)=x/(yz). Compute g(1,1),g(0,1) and...Ch. 7.1 - Let f(x,y,z)=x2ey2+z2. Compute f(1,1,1) and...Ch. 7.1 - Let f(x,y)=xy. Show that f(2+h,3)f(2,3)=3h.Ch. 7.1 - Let f(x,y)=xy. Show that f(2,3+k)f(2,3)=2k.Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...

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All the possible relative maximum or minimum points ( x , y ) of the f ( x , y ) = x 3 + y 2 − 3 x − 8 y + 12 , then use the second-derivative test to determine the nature of f ( x , y ) at each of these points. If the second-derivative test is inconclusive, then state it.

Solution Summary: The author explains how the first-derivative test determines the nature of f(x,y) at each of these points.

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All the possible relative maximum or minimum points (x, y) of the f(x,y)=x3+y2−3x−8y+12, then use the second-derivative test to determine the nature of f(x,y) at each of these points. If the second-derivative test is inconclusive, then state it.

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Chapter 7 Solutions