Find the maximum value of: $$\cfrac{x^4}{x^8+2x^6-4x^4+8x^2+16}$$
I think this means we have to minimize the denominator. But when I am trying to do so, I'm not getting the denominator in terms of a square + constant.
I don't know calculus so please don't give a hint related to that.
I don't know how to start, so hints would be appreciated. This question was given in an exercise based on AM, GM HM inequality, but since they haven't mentioned $x$ is positive, we can't use it I think.
Thanks.
Best Answer
Dividing numerator and denominator by $x^4$, the expression transforms to, $$\dfrac{1}{x^4+2x^2-4+\dfrac{8}{x^2}+\dfrac{16}{x^4}}$$ Now you can apply AM-GM on the terms $x^4,2x^2,\dfrac{8}{x^2},\dfrac{16}{x^4}$ to finish it off.