Question

I am trying to find a bound for the function:
\[\left| \ \frac{ 24x(x^2-1) }{ (1+x^2)^4 } \right|\]
It needs to be less than or equal to M. I can estimate the fraction by making the numerator bigger and denominator smaller to get that the fraction is less than 24, but plotting the graph it is clear that there is a smaller number for the upper bound of this graph. Can anyone share any insight into how I can get closer to the max of the function (which is 4/sqrt(3) for x>0. Cheers

Answer 1

did you try differentiating wrt x and putting it to zero

Answer 2

That's not really the intent of the question. I'm not trying to find the maximum point per se, but rather estimate upper bound of values that the function takes on for x>0

Answer 3

well i was playin around that and i noticed for x>0\[x^3-x<x^2+x^4\]so\[\left| \ \frac{ 24x(x^2-1) }{ (1+x^2)^4 } \right|<24\frac{x^2}{(1+x^2)^3}\]letting \(x=\tan \theta\) for \(0<\theta<\frac{\pi}{2}\)\[\frac{x^2}{(1+x^2)^3}=\sin^2 \theta \ \cos^4 \theta\]setting \(t=\sin^2 \theta\) and \(0<t<1\)\[\sin^2 \theta \ \cos^4 \theta=t(1-t)^2\le\frac{4}{27}\]

Answer 4

so the whole thing\[\left| \ \frac{ 24x(x^2-1) }{ (1+x^2)^4 } \right|<24\frac{4}{27}=\frac{32}{9}\]