Question 1180775
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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.
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<pre>
By analogy with the well known  AM-GM inequality ("Arithmetic Mean - Geometric Mean inequality") for two variables "a" and "b"

    ab <= {{{((a+b)/2)^2}}},         (1)


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

    abc <= {{{((a + b + c)/3)^3}}}.      (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 <= {{{((x^2 + y^2 + z^2)/3)^3}}} = {{{(196/3)^3}}} = {{{196^3/3^3}}}.


Thus the maximum value of  x^2*y^2*z^2,  under the constraint  x^2+y^2+z^2 = 196  is  {{{(196/3)^3}}} = {{{7529536/27}}}.    <U>ANSWER</U>


It is achieved when  x^2 = y^2 = z^2 = {{{196/3}}},  i.e.  x = y = z = +/- {{{sqrt(196/3)}}} = 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.
</pre>

Solved.



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