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Math / Linear Algebra and Its Applications, 4 / Chapter 6 / Problem 6.1.20

Linear Algebra and Its Applications, | 4th Edition | ISBN: 9780030105678 | Authors: Gilbert Strang

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For F1(x,y) = 1 4 x 4 + x 2 y + y 2 and F2(x,y) = x 3 + xy x, find the second derivative

Chapter 6, Problem 6.1.20

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QUESTION:

For F1(x,y) = 1 4 x 4 + x 2 y + y 2 and F2(x,y) = x 3 + xy x, find the second derivative matrices A1 and A2: A = " 2F/ x 2 2F/ x y 2F/ y x 2F/ y 2 # . A1 is positive definite, so F1 is concave up (= convex). Find the minimum point of F1 and the saddle point of F2 (look where first derivatives are zero).

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QUESTION:

For F1(x,y) = 1 4 x 4 + x 2 y + y 2 and F2(x,y) = x 3 + xy x, find the second derivative matrices A1 and A2: A = " 2F/ x 2 2F/ x y 2F/ y x 2F/ y 2 # . A1 is positive definite, so F1 is concave up (= convex). Find the minimum point of F1 and the saddle point of F2 (look where first derivatives are zero).

ANSWER:

Step 1 of 4

Here, given functions are:

Taking second derivative of functions,

So, the matrix .

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