Linear Algebra and Its Applications · Problem 6.1.20

Positive Definite Matrices — Gilbert Strang

Mathematics · positive-definite-matrices-minima-maxima-saddle-points

Problem

For

F_1(x,y) = \frac{1}{4}x^4 + x^2 y + y^2

and

F_2(x,y) = x^3 + xy - x

, find the second derivative matrices

A_1

and

A_2

A = \begin{bmatrix} \partial^2 F/\partial x^2 & \partial^2 F/\partial x \partial y \\ \partial^2 F/\partial y \partial x & \partial^2 F/\partial y^2 \end{bmatrix}.

A_1

is positive definite, so

F_1

is concave up (

=

convex). Find the minimum point of

F_1

and the saddle point of

F_2

(look where first derivatives are zero).