Explanation Given information: The function is f ( x , y ) = 1 x + 1 y + x y and the point is ( 1 , 1 ) . Explanation: Use the Second-Derivative test to determine the nature of the f ( x , y ) = 1 x + 1 y + x y at the point ( 1 , 1 ) . 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 the relative maximum or minimum as follows: If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 , then f ( x , y ) has a relative minimum at ( a , b ) . If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 , then f ( x , y ) has a relative maximum at ( a , b ) . If D ( a , b ) < 0 , then f ( x , y ) has neither a relative maximum nor a relative minimum at ( a , b ) If D ( a , b ) = 0 , then no conclusion can be drawn from this test. Differentiate f ( x , y ) = 1 x + 1 y + x y partially twice with respect to x and y , it becomes ∂ f ∂ x = ∂ ∂ x ( 1 x + 1 y + x y ) ⇒ ∂ f ∂ x = − 1 x 2 + y ∂ ∂ x ( ∂ f ∂ x ) = ∂ ∂ x ( − 1 x 2 + y ) ⇒ ∂ 2 f ∂ x 2 = 2 x 3 ∂ f ∂ y = ∂ ∂ y ( 1 x + 1 y + x y ) ⇒ ∂ f ∂ y = − 1 y 2 + x ∂ ∂ y ( ∂ f ∂ y ) = ∂ ∂ y ( − 1 y 2 + x ) ⇒ ∂ 2 f ∂ y 2 = 2 y 3 Now, differentiate ∂ f ∂ y = − 1 y 2 + x with respect to x by treating y as constant

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Chapter 7.3, Problem 24E

To determine

The nature of f(x, y)=1x+1y+xy at the point (1, 1) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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Chapter 7.3, Problem 23E

Chapter 7.3, Problem 25E

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Chapter 7 Solutions

Pearson eText for Calculus & Its Applications -- Instant Access (Pearson+)

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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The nature of f ( x , y ) = 1 x + 1 y + x y at the point ( 1 , 1 ) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

Solution Summary: The author explains the second-derivative test to determine the nature of f(x,y)=1x+

The nature of f(x, y)=1x+1y+xy at the point (1, 1) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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Chapter 7 Solutions