We will searching for stationary points, qualifying then as local maxima/minima.
First we will transform the maxima/minima with inequality restrictions into an equivalent maxima/minima problem but now with equality restrictions.
To do that we will introduce the so called slack variables s_1s1 and s_2s2 such that the problem will read.
Maximize/minimize f(x,y) = x^2 + 3 x y + 9 y^2f(x,y)=x2+3xy+9y2
constrained to
{
(g_1(x,y,s_1)=x + 3 y - s_1^2=0),
(g_2(x,y,s_2)=x + 3 y + s_2^2 - 2=0)
:}
The lagrangian is given by
L(x,y,s_1,s_2,lambda_1,lambda_2) = f(x,y)+lambda_1 g_1(x,y,s_1)+lambda_2g_2(x,y,s_2)
The condition for stationary points is
grad L(x,y,s_1,s_2,lambda_1,lambda_2)=vec 0
so we get the conditions
{
(lambda_1 + lambda_2 + 2 x + 3 y = 0),
(3 lambda_1 + 3 lambda_2 + 3 x + 18 y = 0),
( -s_1^2 + x + 3 y = 0),
( -2 lambda_1 s_1 = 0),
(-2 + s_2^2 + x + 3 y = 0),
(2 lambda_2 s_2 = 0)
:}
Solving for {x,y,s_1,s_2,lambda_1,lambda_2} we have
{(x = 0., y = 0., lambda_1 = 0., s_1 = 0., lambda_2 = 0.,
s_2 = 1.41421), (x = 1., y = 0.333333, lambda_1 = 0., s_1 = -1.41421,
lambda_2 = -3., s_2 = 0.)
:}
so we have two points p_1={0,0} and p_2 = {1,0.333333}
Point p_2 activates restriction g_1(x,y,0)=0,{lambda_1 ne 0, s_1 = 0}
p_1 is qualified with f(x,y)
and
p_2 is qualified with f_{g_2}(x) =x(x-2)+4
Computing
grad f(0,0) = 0
and
"Eigenvalues"(grad^2 f(0,0))={18.544, 1.456}
we conclude that p_1 local minimum point.
Analogously for p_2
d/(dx)(f_{g_2}(0)) = 0
and
d^2/(dx^2)(f_{g_2}(0)) = 2
so p_1,p_2 are local minima points
