Show that for both of the following two problems, P = (r, h) is a Lagrange critical point if h = 2r:
• Minimize surface area S for fixed volume V.
• Maximize volume V for fixed surface area S .

Then use the contour plots in Figure 18 to explain why S has a minimum for fixed V but no maximum and, similarly, V has a maximum for fixed S but no minimum.

Transcribed Image Text:

h Increasing S Level curves of S Critical point P = (r, h) Level curve of V Increasing V 1