Lagrange Multipliers

Question

Find all of the extrema of g(x,y,z)=xyz on the surface z=e^(-x^2-y^2).

Answer 1

Introduce a Legrange multiplier, λ.    The function to be studied is then:

  h(x,y,z) =  xyz + λ (z - exp(-x²-y²))

   Denote the partial  derivatives with subscripts ( for example  h_x is the partial derivative of h wrt x)

Then the conditions for extrema  are:   h_x =0, h_y =0 and h_z = 0      These three equations work out to:

 

    yz + λ (2x exp(-x² -y²))  = 0

    xz  +λ(2y exp(-x² -y²) )= 0

    xy + λ  =0 

 

   The last of these is easily solved for λ,  giving  λ =  -xy.  Substituting this into the first two, eliminating z using

the constraint equation z = exp(-x²-y²),  and factoring out the common factor of exp(-x²-y²)  gives:

 

   y - 2 x² y  = 0   and x - 2 x y²  =0

 

This set of two nonlinear equations has 5 solutions:   

  The obvious one is  x = y = 0.   

  The other four can be obtained by multiplying the first one by y and the second one by x.   This leads to

   x² = y²     ,  Thus y =  ± x.    Plugging this into the first one yields   x = ± sqrt(1/2).   Thus in ordered triplet

notation  (x,y,z),   the five extrema are:

 

    (0,0,1)

    ( sqrt(1/2), sqrt(1/2),  exp(-1)  )

    ( sqrt(1/2), -sqrt(1/2), exp(-1) )
    ( -sqrt(1/2), sqrt(1/2), exp(-1/ )
    ( -sqrt(1/2), -sqrt(1/2), exp(-1) )