Lagrange multiplier

Question

The atmospheric re-entry temperature on the surface of a space capsule represented by the
ellipsoid 4x2 + y 2 + 4z 2 = 16 is given by T = 8x2 + 4yz -16z + 600 (°C) . The material from
which the capsule is made can only withstand a temperature of 650°C. Will the occupants of
the space-craft survive re-entry

Answer 1

Tx=16x  =¥Gx                                            
Ty=4z    =¥Gy                                                      
Tz=-16   =¥Gz
let g(x,y,z)=4x^2+y^2+4z^2-16
Gx=8x Gy=2y Gz=8z
hence ¥=2 x=1 y=-1 z=-1
G(1,-1,-1)=628 They will survive