SOLUTION: Find the maximum values of the function f(x,y,z)=x^2y^2z^2 subject to the constraint x^2+y^2+z^2=196. Maximum value is: My anwer is = 27/7529536 but it's wrong.

By analogy with the well known  AM-GM inequality ("Arithmetic Mean - Geometric Mean inequality") for two variables "a" and "b"

    ab <= ,         (1)


there is AM-GM inequality for three variables "a", "b" and "c"

    abc <= .      (2)


    Inequalities (1) and (2) are valid for any two and three variables, respectively, that are real non-negative numbers.


    In inequalities (1) and (2), equalities are achieved if and only if  a = b  (for (1))  or  a = b = c (for (2)).



Apply inequality (2), taking  

    a = x^2,  b = y^2,  c = z^2.


You will get

    x^2*y^2*z^2 <=  =  = .


Thus the maximum value of  x^2*y^2*z^2,  under the constraint  x^2+y^2+z^2 = 196  is   = .    ANSWER


It is achieved when  x^2 = y^2 = z^2 = ,  i.e.  x = y = z = +/-  = 8.082904  (rounded).


In all, there are 8 points on the 3D sphere surface  x^2 + y^ + z^2 = 196,
where the maximum value of x^2*y^2*z^2 is achieved - one such point in each octant.