Question 1209923
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Let  x, y, and z be real numbers. If x^2 + y^2 + z^2 = 1, then find the maximum value of
3x + 4y + 5z + x^3 + {{{(4x^2*y)/z}}} + {{{(z^5)/(xy^2)}}}.
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            @CPhill finds it difficult to give a definitive answer in his post.



Meanwhile,  the answer to this question is very simple:  under given conditions, 
the given function/expression has  NO  maximum.


It is because the term   {{{(4x^2*y)/z}}}   of the expression has variable  z  in the denominator.



        Take  (x,y,z)  in vicinity of   ({{{sqrt(2)/2}}},{{{sqrt(2)/2}}},{{{0}}}),    so that    x^2 + y^2 + z^2 = 1   is valid, 


                  and let  z  goes to zero from the positive side.



Then the term   {{{(4x^2*y)/z}}}   tends to positive infinity,
and with this term, the whole expression tends to positive infinity.



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So the answer in the post by @CPhill is empty.