Consider gxy ax2 y2cy where a 0 a Find a unique critical

Consider g(x,y) = ax^2 + y2^-cy, where a > 0. (a) Find a unique critical point (x,y) of g. (b) Denote the critical point as (x,y). For any (x,y), expand g(x,y) - g(x,y) as a sum of linear term and a quadratic term using the Taylor\'s theorem. (c) Prove using the result in b that (x,y) is a global minimizer.

Solution

given g(x,y)= ax^2 + y^2 - cy

on differentiating with respect to x g\'(x) = ax

to find critical points this tends to zero

ax=0

x=0 this is one critical point

g\'(y) = Y - C

this tends to zero

Y = C

on using taylors theorem where

f(x) = f(a) + f\'(a) (X-a) + f\'\'(a) (x-a)^2/2

= ax^2 + y^2 - cy + ax + y -c + a /2

= a ( x^2 + x + 1/2) + y^2 + y -c(y+1)

as y=c

f(x) =   a ( x^2 + x + 1/2) + C^2/2 + c - c^2 -c

=    a ( x^2 + x + 1/2) - c^2/2

at x=0

f(x) = a/2 - c^2/2

which equals to zero

c^2 = a

for different values of e there will be different critical points

g(x,y) is a global minimizer