Question 1209940
.
The function f has the following properties:
* f(x) is defined for x > 0
* f(x) > 0 for all x > 0
* f(x - y) = {{{sqrt(f(xy) + 4x + 4y + 8)}}} for all x > y > 0.

Determine f(1).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



        In his post,  Edwin introduced right and nice idea to work with numbers  x  and  y  such that


                x-y = 1,   x*y = 1.


        It leads directly to the solution,  but on the way,  in the implementation process, 

        Edwin made arithmetic errors that lead him to wrong answer.


        In this my post, I copied  Edwin's calculations,  but fixed them to repair that error.


        See my solution below.   At the end,  I checked my answer to prove its validity.



<pre>
{{{f(x-y) = sqrt(f(x*y)+4*x+4*y+8)}}}   <<<---===  notice that Edwin mistakenly writes "-8" in this place

{{{f(x-y) = sqrt(f(x*y)+4*x+4*y+8)}}}   <<<---===  notice that Edwin mistakenly writes "-8" in this place

If we can find a case of x and y where {{{f(x-y)=f(x*y)=f(1)}}}, then we could
solve for f(1).  That would require  

{{{x-y=x*y=1}}}

{{{y=1/x}}}, {{{x-1/x=1}}}

{{{x-1/x=1}}}
{{{x^2-1=x}}}
{{{x^2-x-1=0}}}
{{{x=(1 +- sqrt(1+4))/2}}} <--we can only use the + sign
{{{x=(1+sqrt(5))/2}}}

Incidentally, that, or its reciprocal, is the golden ratio, famous in 
historical architecture. Let's call it {{{G=(1+sqrt(5))/2}}}.

{{{G=(1+sqrt(5))/2}}} and {{{G-1/G=1}}}

{{{f(G-1/G) = sqrt(f(G*(1/G))+4*G+4*expr(1/G)+8)}}} <<<---===  notice that Edwin mistakenly lost  "+8"  at this place.
                                                 This place was the Edwin's fatal error.
                                                 After fixing it, my numbers and my calculations are different from that by Edwin.

f(1) = {{{sqrt(f(1)+4*(G+1/G)+8)}}}

Since {{{G-1/G=1}}}, {{{1/G=G-1}}}

f(1) = sqrt(f(1) + 4*(G+G-1) + 8)}}}

f(1) = sqrt(f(1) + 4*(2G-1) + 8)}}}

f(1) = sqrt(f(1) + 8G + 4)}}}

(f(1))^2 = f(1) + 8G + 4

(f(1))^2 - f(1) - (8G + 4) = 0

Solve for f(1) using the quadratic formula

f(1) = {{{(1 +- sqrt(1+4(8G+4)))/2}}}  <--- we can only use +

f(1) = {{{(1 + sqrt(1+32G+16))/2}}}

f(1) = {{{(1 + sqrt(32G+17))/2}}}

Substituting  G = {{{(1+sqrt(5))/2}}}, and doing some algebra,
it simplifies to

f(1) = {{{(1+sqrt(33+16sqrt(5)))/2}}}


<U>ANSWER</U>.  f(1) = 4.646597631  approximately.


                      <U>CHECK</U>


I will check that the original equation is correct

    f(1) = {{{sqrt(f(1)+4*x+4*y+8)}}}.    (*)


Here left side  f(1)  is 4.646597631;

    x = {{{(1+sqrt(5))/2}}} = 1.618033989;  

    y = {{{1/x}}} = {{{1/1.618033989}}} = 0.618033989  (the same as y = x - 1);


so, the right side of (*) is

    {{{sqrt(4.646597631 + 4*1.618033989 + 4*0.618033989+8)}}} = 4.646597631,

which is PRECISELY the same as the left side.


So, my answer is correct and is confirmed.   The CHECK is completed.
</pre>

Solved.



/////////////////////////////////



Do not consider the post by @CPhill seriously.


This post is simply a blatant attempt to deceive a reader 
by presenting his fake work as a solution to a mathematical problem.


It does not contain a proper mathematical deducing from the beginning to the end.


I am writing this in order to express / (to describe) the true state of affairs in right words.



\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . 



Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.


The artificial intelligence is like a baby now. It is in the experimental stage 
of development and can make mistakes and produce nonsense without any embarrassment.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.



This time, again, &nbsp;it made an error.



Although the @CPhill' solution are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.


Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental
and can make errors/mistakes.


All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.


Every time, &nbsp;@CPhill embarrassed to tell the truth.

But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.



And the last my comment.


When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, 
until the specialists and experts will check and confirm their correctness.


Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.