%   Problem 14.8.16:  Max/min with two constraints 
% 
 


o = [0,0,0];   % origin will be shifted otherwise we won't see the arrows inside our ellipsoid

 arrow3(o, [1,0,0],'yellow');
 arrow3(o, [0,1,0],'yellow');
 arrow3(o, [0,0,1],'magenta');    % this plots the i,j,k, vectors
 
 ymin = 0;
 ymax = 4;       % plotting window
 
 
   %%  Example 1. Elliptic paraboloid
   %
   %   its equation is  z = x^2 + y^2 
  
 % in fact, even to plot the cylinder, we need to use these  
 dtheta = pi/48;  
 r = 1;
 
 [theta,Y] = meshgrid(0:dtheta:2*pi, ymin:0.2:ymax); 
   X = r*cos(theta);
   Z = r*sin(theta)/sqrt(2);
          surf(X,Y,Z,'FaceColor',[1 1 0.8]);        % surface plotting
 
     % now plot x + y -z = 0   plane 
     
   dy= 0.2;
   [Y,Z ] = meshgrid(-1:dy:ymax, -1:dy:1);
   X = Z - Y;
      surf(X,Y,Z,'FaceColor',[1 0.8 0.8]);   
     
     %% plot their intersection
   t = 0:dtheta:2*pi;
   x = cos(t);
   z = sin(t)/sqrt(2);
   y = z - x;
     plot3(x,y,z,'r','LineWidth',3)
      
      
     %% now plot the level set for the f(x,y,z) = 3x - y - 3z = - 12/sqrt(6)
     
   X2 = (Y + 3*Z - 12/sqrt(6))/3;
        surf(X2,Y,Z,'FaceColor',[0.8 0.8 1]);   
     
     
     
      
      
     t = 0:0.2:8;
     theta = t;
     
   x = r*cos(theta);
   y = r*sin(theta);
   z = t/2; 
   
     hold on
     plot3(x,y,z,...
                'LineWidth',2);
     hold off