Can the Maximum of Rational Functions Be Split into Separate Maxima?

[japam]

Suppose 2 cuadratic functions: ax^2+bx+c, dx^2+ex+f. Suppose that the first one is upside with its minimum above the x line reference, and the second one is downside with its maximum above the x reference, and suppose that the two functions intersect at two points that pass through straight line gx+h.
My question is
¿could the maximum of (dx^2+ex+f)/(ax^2+bx+c), be splitted as the
max( (dx^2+ex+f)/(gx+h))*max((gx+h)/(ax^2+bx+c))?

I don't know how to put drawings here, but i hope the argument has been clear to understand
I was tryng some numeric examples in my pc and the result was positive, but i don't know what is the general proof. Thanks for your comments

 

[fresh_42]

No, this cannot be done. The maxima don't even need to exist. Test it on $-x^2+4\, , \,3x^2$ with $y=3$ as straight. And you must not multiply infinities.

 

[WWGD]

Suppose 2 cuadratic functions: ax^2+bx+c, dx^2+ex+f. Suppose that the first one is upside with its minimum above the x line reference, and the second one is downside with its maximum above the x reference, and suppose that the two functions intersect at two points that pass through straight line gx+h.
My question is
¿could the maximum of (dx^2+ex+f)/(ax^2+bx+c), be splitted as the
max( (dx^2+ex+f)/(gx+h))*max((gx+h)/(ax^2+bx+c))?

I don't know how to put drawings here, but i hope the argument has been clear to understand
I was tryng some numeric examples in my pc and the result was positive, but i don't know what is the general proof. Thanks for your comments

That would only work if max was reached for the same values of x, e.g., Max $sinx \neq$ (Max ${xsinx})$ (Max$\frac {1}{x})$ ( for one, last expression is unbounded, while $sinx$ has $1$ as its max.