How do you minimize and maximize #f(x,y)=x^2+y^3# constrained to #0<x+3xy<4#?

Introducing the so called slack variables #s_1, s_2# the optimization problem is transformed into an equivalent one

Find local minima, maxima of

#f(x,y)= x^2+y^3#

subject to

#g_1(x,y,s_1) = x + 3 x y - s_1^2 = 0#
#g_2(x.y,s_2)=x+3 x y + s_2^2-4=0#

The lagrangian is

#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)#

#L# is analytical so the stationary points include the relative maxima and minima.

The determination of stationary points is done solving for #x,y,s_1,lambda_1,s_2,lambda_2# the system of equations given by

#grad L(x,y,s_1,lambda_1,s_2,lambda_2) = vec 0#

or

#{(2 x + lambda_1 (1 + 3 y) + lambda_2 (1 + 3 y)=0), (3 lambda_1 x + 3 lambda_2 x + 3 y^2=0), ( -s_1^2 + x + 3 x y=0), (-2 lambda_1 s_1=0), (-4 + s_2^2 + x + 3 x y=0) ,( 2 lambda_2 s_2=0) :} #

Solving we get

#(x=0., y = 0., lambda_1 = 0., s_1 = 0., lambda_2= 0., s_2= 2.)#

and

# (x= 1.12872, y= 0.847942, lambda_1=0., s_1 = 2., lambda_2 = -0.637008, s_2 = 0.)#

Both points are at the boundaries of #g_1(x,y,0)=0# and #g_2(x,y,0)=0# respectively.

Their qualification must be done with #f_(g_1)# and #f_{g_2}# respectively. So,

#f_{g_1}(x) = x^2-1/27#
#f_{g_2}(x) = -(x-4)^3/(27 x^3) + x^2#

#d^2/(dx^2)f_{g_1}(0)=2# qualifying this point as a local minimum
#d^2/(dx^2)f_{g_2}(1.12872) =11.5726# qualifying this point as a local minimum also

Attached a figure with a contour mapping with the found points