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 sqrt[x y + y]f(x,y)=x√xy+y
constrained to
{
(g_1(x,y,s_1)=x y - y^2 - s^2=0),
(g_2(x,y,s_2)=x y - y^2 + s_2^2 - 5=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 y + lambda_2 y + (x y)/(2 sqrt[y + x y]) + sqrt[y + x y] = 0),
(lambda_1 (x - 2 y) + lambda_2 (x - 2 y) + (x (1 + x))/(2 sqrt[y + x y]) = 0),
( -s_1^2 + x y - y^2 = 0),
(-2 lambda_1 s_1 = 0),
(-5 + s_2^2 + x y - y^2 = 0),
(2 lambda_2 s_2 = 0)
:}
Solving for {x,y,s_1,s_2,lambda_1,lambda_2} we have
{(x = -4.69709, y = -1.63044, lambda_1 = 0., s_1 = -2.23607,lambda_2 = 2.4624,s_2 = 0.),
(x = 4.78125, y = 1.54499, lambda_1 = 0., s_1 = -2.23607,
lambda_2 = -2.73431, s_2 = 0.)
:}
so we have two points
p_1={-4.69709, -1.63044}
and
p_2 = { 4.78125, 1.54499}
Point p_1 activates restriction g_2(x,y,0)=0,{lambda_2 ne 0, s_2 = 0}
Point p_2 activates restriction g_2(x,y,0)=0,{lambda_2 ne 0, s_2 = 0}
p_1 is qualified with f_{g_2}(x)=(x sqrt[(1 + x) (x + sqrt[ x^2-20])])/sqrt[2]
and
p_2 is qualified with f_{g_2}(x)
Computing
d/(dx)(f_{g_2}(-4.69709)) = 0
and
d^2/(dx^2)(f_{g_2}(-4.69709)) = -8.19783
we conclude that p_1 local maximum point.
Analogously for p_2
d/(dx)(f_{g_2}(0)) = 4.78125
and
d^2/(dx^2)(f_{g_2}( 4.78125)) = 6.22258
so p_1,p_2 are local maximum and minimum points
