Question 1208298
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if (x/y) + (y/x) = 1 , then y'' = (y/x) (True or False)
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<pre>
Let's investigate which  (or what)  function y(x) can be.

We have

    {{{x/y)}}} + {{{y/x}}} = 1.


Let  z = {{{y/x}}}.  Then

    {{{1/z}}} + z = 1,

    1 + z^2 = z,

    z^2 - z + 1 = 0,

    {{{z[1,2]}}} = {{{(1 +- sqrt(1 - 4))/2}}} = {{{(1 +- i*sqrt(3))/2}}}.


So,  {{{y/x}}}  has the constant value  {{{z[1]}}}  or  {{{z[2]}}}.  Hence,


    y = z*x,  where  z is one of the two complex numbers  z = {{{z[1]}}} = {{{(1 + i*sqrt(3))/2}}}  or  z = {{{z[2]}}} = {{{(1 - i*sqrt(3))/2}}}.


It implies

    y' = z = const  and then  y'' = 0.


Thus we have to compare, from one side,  y'' = 0  and, from the other side,  {{{y/x}}} = {{{z[1]}}} or {{{z[2]}}},  what are not zero.


        So, the answer to the problem's question is  {{{highlight(highlight(FALSE))}}}.


But an interesting fact, which is worth to be noticed, is that  y' = z = {{{y/x}}}  has one of the two possible constant values

{{{(1 + i*sqrt(3))/2}}}  or  {{{(1 - i*sqrt(3))/2}}}.
</pre>

Solved, answered and explained.