Lagrange Multipliers Finding largest and smallest value of T(x,y) = 8 - x^2 - y^2 on the ellipse x^2 + xy + y^2 = 9/4 Temperature distribution on a rectangular plate -2.5 < x < 2.5, -2.5 < y < 2.5 containing this ellipse restart; with(plots): T := 8 - x^2 - y^2: Tplot := contourplot( T, x=-2.5..2.5, y=-2.5..2.5, filled=true, coloring=[blue,red] ): display( Tplot, axes=boxed ); The hottest point on this plate is near the center and the coldest points are near the four corners of the rectangle. We are interested in the more subtle question - we have a wire in the shape of the ellipse . If the temperature at the point (x,y) on the wire is given by the same function determine the hottest and the coldest point on the wire. We draw the temperature distribution T(x,y) on the ellipse (wire). ellipse := implicitplot( x^2 + x*y + y^2 = 9/4, x=-2..2, y=-2..2, color=white,thickness=2 ): display( {Tplot, ellipse}, axes=boxed ); Which point on the ellipse is the hottest and which is the coldest - remember red is hot and blue is cold. Perhaps you have concluded that the hottest points (click the graph there) on the ellipse are near (-0.8,-0.8) and (0.8,0.8), and the coldest points on the ellipse are (click there) near (-1.5,1.5) and (1.5,-1.5). However, this procedure is not satisfactory, since it only gives the approximate location. To determine the precise points we have to find a mathematical characterization of these points. That is what we do below. We know that the max temp on the RECTANGULAR PLATE is 8 and the minimum temperature is . So the max and min temp on the ellipse is somewhere in this range. Let us find all the points on the ellipse where the temp is 0. We do this by drawing the contour curve T(x,y)=0 (color = green) contour := implicitplot( T = 0, x=-2.5..2.5, y=-2.5..2.5, thickness=2, color=green ): display( {Tplot, ellipse, contour} ); There seem to be no points on the ellipse where the temperature is zero. Further every point on the ellipse has a temperature higher than zero. Let us find all the points where the temperature is 4. contour := implicitplot( T = 4, x=-2.5..2.5, y=-2.5..2.5, thickness=2, color=green ): display( {Tplot, ellipse, contour} ); There are four points on the ellipse where the temperature is 4 units. So the minimum temperaure on the ellipse is between 0 and 4 units. Let us find it with a little more accuracy. We try T=2. contour := implicitplot( T = 2, x=-2.5..2.5, y=-2.5..2.5, thickness=2, color=green ): display( {Tplot, ellipse, contour} ); There are no points on the ellipse where T=2. Continue in this fashion (just modify the previous command line) and find the max and the min temperature on the ellipse. Also find the coordinates of the point where this is attained. However, we still have not come up with a mathematical characterization of the point where the max and min are attained - or do we have a characterization now?