Question

Given a two-variable function f(x, y), if P(x0,yo) is a critical point, then the behavior of f around P can be approximated by its second order terms according to Taylor series, that is, f(x,y) = f(P) + F(x – xo)?H (x, y) , where H(x, y) = fyy(P)(=%)2 + 2 fxy(P) (?=%) + fxx(P). (a). If H(x, y) > 0 for all x,y, is P a local max, local min or saddle point? (b). Let s = (4=90). Then, H(x, y) = fyy(P) s2 + 2fxy(P)s + fxx(P), a quardric function in term of s. Use Problem.1 to get the discriminant for local max,local min and saddle point.

Answer 1

When the eigenvalue of a matrix is both positive, then this is positive definite, ; when both negative, then negative definite;

when one is positive and one is negative, then indefinite. We use this fact to find the conditions on the discriminant D(x,y).

See in detail:

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foy fry tootifm a ve Iw i Cade >o. - Lo aminima. -ve de tnite t Tn t Cefy) )< , (ii in detinite, when D) o add k point a Th th diseriminant is 0y,*

foy fry tootifm a ve Iw i Cade >o. - Lo aminima. -ve de tnite t Tn t Cefy) )< , (ii in detinite, when D) o add k point a Th th diseriminant is 0y,*