The function F ( x ) is defined by the following identity:
F ( ( F ( x ) + x ) k ) = ( F ( x ) + x ) 2 − x
The value of F ( 1 ) is such that a finite number of possible numerical values of F ′ ( 1 ) can be determined solely from the above information. The maximum value of k such that F ′ ( 1 ) is an integer can be expressed as b a , where a and b are coprime integers. What is the value of a + b ?
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I have several concerns
1) What is the domain of your function?
2) Why must F ( x ) be differentiable? Arguably, your question makes the implicit assumption that F ′ ( 1 ) exists, so this could be minor.
3) Why do we have F ′ ( ( F ( 1 ) + 1 ) k ) = F ′ ( 1 ) ⇒ ( F ( 1 ) + 1 ) k = 1 ? The backward direction is true, but the forward direction only follows if F ′ ( x ) is a surjective function (which has yet to be shown).
4) From ( F ( 1 ) + 1 ) k = 1 , why can't we have F ( 1 ) = − 2 and k be even?
5) Does such a function exist? You merely showed properties that it must have, but have not shown that there is a function which satisfies the conditions.
6) Given that your k is not an integer, do we need to be concerned if F ( x ) + x < 0 ?
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By differentiating both sides and applying the chain rule (twice in the case of the left side), we get the following:
k ( F ( x ) + x ) k − 1 ( F ′ ( x ) + 1 ) F ′ ( ( F ( x ) + x ) k ) = 2 ( F ( x ) + x ) ( F ′ ( x ) + 1 ) − 1
Substituting 1 in for x , we then get
k ( F ( 1 ) + 1 ) k − 1 ( F ′ ( 1 ) + 1 ) F ′ ( ( F ( 1 ) + 1 ) k ) = 2 ( F ( 1 ) + 1 ) ( F ′ ( 1 ) + 1 ) − 1
We must choose F ( 1 ) such that F ′ ( ( F ( 1 ) + 1 ) k ) = F ′ ( 1 ) ⇒ ( F ( 1 ) + 1 ) k = 1 , which gives F ( 1 ) = 0 . Thus, our equation above becomes k F ′ ( 1 ) ( F ′ ( 1 ) + 1 ) = 2 ( F ′ ( 1 ) + 1 ) − 1 , the solution to which is F ′ ( 1 ) = 2 k 2 − k ± k 2 + 4 . For the case F ′ ( 1 ) = 2 k 2 − k − k 2 + 4 , F ′ ( 1 ) is asymptotic to 0 as k approaches − ∞ and − 1 as k approaches ∞ . Thus, for this case, F ′ ( 1 ) will never be an integer. For the case F ′ ( 1 ) = 2 k 2 − k + k 2 + 4 , F ′ ( 1 ) is positive for positive k and asymptotic to 0 as k approaches ∞ . Thus, the largest value of k such that F ′ ( 1 ) is an integer is the largest value of k where F ′ ( 1 ) = 1 . 2 k 2 − k + k 2 + 4 = 1 ⇒ k 2 + 4 = ( 3 k − 2 ) 2 ⇒ 2 k 2 − 3 k = 0 ⇒ k = 2 3 , so a + b = 5 .