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_1# and #s_2# such that the problem will read.

Maximize/minimize #f(x,y) = x sqrt[x y + 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