# STEP question.Watch

#1
The question goes like this:
Let f(x)= ax - x³/(1+x²)
Show that "if a => 9/8, then f`(x) => 0 for all x"

I seem to have done it, but the method is sorta different from the one given in the solution (it's in some book...). And I want to ask you guys if you think it's satisfactory.

Okay, the question's to do with f`(x), so I differentiate:
a - [(1+x²)(3x²)-x³(2x)] /(1+x²)²
f`(x)= a - (3x²+x^4) /(1+x²)²
Let everything behind the minus sign be "horrible function".
If f`(x) => 0, then the horrible function must be less than 'a'. (Or equal to it)

The maximum value of the horrible fraction can be found by differentiating it and setting the result =0, and it turns out to occur when x=sqrt(3).
Horrible function(sqrt(3)) gives 9/8, and so if f`(x) is to be =>0, then a=> 9/8, which is the maximum value of the horrible function.

Would that be satisfactory to prove the original question?

(Also, am I allowed to break a little rule and ask if there's an archive of these papers anywhere? XD)

Thanks
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12 years ago
#2
Archive.

Did you confirm that 9/8 gives a maximum rather than a minimum?

Otherwise it would probably be acceptable (although, alas, I'm not a STEP examiner) - but differentiating to find the maximum of the 'horrible fraction' is a rather drawn out way of doing it, so a quicker way would have been better (as Siklos does in the booklet). Also, you seem to have shown 'if and only if' - which is more than necessary. In questions with "Prove P if Q" or "Prove P only if Q" (or questions about necessary and sufficient conditions), you have to make sure you prove the correct implication - also, having looked at the examiner's reports for 2001-2005, a lot of candidates seem to forget, for example, to prove "P only if Q" in "show P if and only if Q" questions.

Note that if you want to show , say, is greater than or equal to zero, you could try to show (e.g. by completing the square) that (where c is non-negative), since squares have the required property.
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12 years ago
#3
I haven't read your argument thoroughly, but from the wording I think you've only shown "if f'(x)>=0 then a >= 9/8", whereas you want to show the other way around. It seems like your argument is easily reversed, but be careful with wording and make sure you prove implications the right way around.
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