%% Using Lagrange Multipliers to Solve a Constrained Max-Min Problem

%% Space probe

%%
% A space probe in the shape of the ellipsoid 
% 
% $$4x^2+y^2+4z^2=16$$
% 
% enters the earth's atmosphere and its surface begins to heat.  After 1
% hour the temperature at the point (x,y,z) on the probe's surface is
%
% $$T(x,y,z) = 8x^2+4yz-16z+600. $$
%
% Find the hottest point on the probe's surface.
%
% *Step1*  Write the constraint as g = 0.  The function g is
syms x y z
g = 4*x^2+y^2+4*z^2-16

%%
% *Step 2* The extreme values of T on the probe's surface occur where the
% gradients of T and g are parallel.  We set up the equations

T = 8*x^2+4*y*z-16*z+600

%%

gradg = jacobian(g,[x,y,z])
gradT = jacobian(T,[x,y,z])

%% 
% We need to solve the equations
%
% $$ \nabla T \,\mbox{--}\,\lambda \nabla g = 0, \ g=0. $$
%
% *Step 3* We solve the equations.
syms lam
[lsol,xsol, ysol, zsol] = solve(gradT(1)-lam*gradg(1),gradT(2)-lam*gradg(2),gradT(3)-lam*gradg(3),g);
[xsol,ysol,zsol,lsol]

%% 
% *Step 4* Plug the resulting values for (x,y,z) into T and look to see
% where the maximum occurs.
% First we turn T into a function
Tfun = inline(vectorize(T))
double([xsol,ysol,zsol, Tfun(xsol,ysol,zsol)])

%%
% The hottest points on the probe's surface are
%
% $$\left(\pm \frac 4 3 , \,\mbox{--}\,\frac 4 3 ,\,\mbox{--}\,\frac 4 3 \right).$$