Module 13 : Maxima, Minima and Saddle Points, Constrained maxima and minima
Lecture 39 : Constrained maxima / minima [Section 39.2]
(3)
Using the Lagrange's method of multiplier, show that the minimum value of is and is attained at
and the maximum value of subject to the constraint that lies on the unit sphere is and it is attained at . Using these, deduce the A.M.-G.M. inequality: for three nonnegative real numbers ,
(4)
A space probe in the shape of the ellipsoid
enters the earth's atmosphere and its surface begins to heat. After one hour, the temperature at the point on the surface of the probe is given by
Find the hottest point on the surface of the probe.
is 
and is attained at 
subject to the constraint that 
lies on the unit sphere is 
and it is attained at 
. Using these, deduce the A.M.-G.M. inequality: for three nonnegative real numbers 
, 
on the surface of the probe is given by 
