Explanation Given information: The function is f ( x , y ) = 2 x 2 − x 4 − y 2 and the points ( − 1 , 0 ) , ( 0 , 0 ) and ( 1 , 0 ) . Explanation: Use the Second-Derivative test to determine the nature of the f ( x , y ) = 2 x 2 − x 4 − y 2 at the points ( − 1 , 0 ) , ( 0 , 0 ) and ( 1 , 0 ) . 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 ) . 4. If D ( a , b ) = 0 , then no conclusion can be drawn from this test. Differentiate f ( x , y ) = 2 x 2 − x 4 − y 2 partially twice with respect to x and y , it becomes ∂ f ∂ x = ∂ ∂ x ( 2 x 2 − x 4 − y 2 ) ⇒ ∂ f ∂ x = 4 x − 4 x 3 ∂ ∂ x ( ∂ f ∂ x ) = ∂ ∂ x ( 4 x − 4 x 3 ) ⇒ ∂ 2 f ∂ x 2 = 4 − 12 x 2 ∂ f ∂ y = ∂ ∂ y ( 2 x 2 − x 4 − y 2 ) ⇒ ∂ f ∂ y = − 2 y ∂ ∂ y ( ∂ f ∂ y ) = ∂ ∂ y ( − 2 y ) ⇒ ∂ 2 f ∂ y 2 = − 2 Now, differentiate ∂ f ∂ y = − 2 y with respect to x by treating y as constant. ∂ ∂ x ( ∂ f ∂ y ) = ∂ ∂ x ( − 2 y ) ⇒ ∂ 2 f ∂ x ∂ y = 0 Substitute these values of ∂ 2 f ∂ x 2 = 4 − 12 x 2 , ∂ 2 f ∂ y 2 = − 2 , and ∂ 2 f ∂ x ∂ y = 0 in the formula D ( x , y ) = ∂ 2 f ∂ x 2 × ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2

15th Edition

ISBN: 9780137590896

Author: Larry J. Goldstein; David C. Lay; David I. Schneider; Nakhle H. Asmar; William Edward Tavernetti

Publisher: Pearson Education (US)

See similar textbooks

Question

Chapter 7.3, Problem 21E

To determine

The nature of f(x, y)=2x2−x4−y2 at given points (−1, 0),(1, 0), and (1, 0) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.3, Problem 20E

Chapter 7.3, Problem 22E

Students have asked these similar questions

Given that 4−4i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. f(x)=x4−5x3−2x2+176x−320

eliminate the parameter to find the cartesian equation of the curve and sketch the graph. On the graph show the direction it takes and the initial and terminal point. Please draw by hand and show how you got to each step

Chapter 7 Solutions

Calculus & Its Applications

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...

Knowledge Booster

The nature of f ( x , y ) = 2 x 2 − x 4 − y 2 at given points ( − 1 , 0 ) , ( 1 , 0 ) , and ( 1 , 0 ) 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)=2x2-4-

The nature of f(x, y)=2x2−x4−y2 at given points (−1, 0),(1, 0), and (1, 0) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

Want to see the full answer?

Chapter 7 Solutions