Find maximum of quotient of functions

Question

To maximize $f(x)=\frac{g(x)}{h(x)}$ on all of $R$, I think I have found a method.

First, find the point at which $g(x)$ is maximized. Call it $a$.

Second, find the point at which $h(x)$ is minimized. Call it $b$.

Finally, compute $f(a)$ and $f(b)$. Whichever one is larger is the maximum of $f(x)$.

Can anyone find a proof or disproof of my approach? I've tested it for several functions and it seems to work.

score 1 · Answer 1 · 2015-09-09T01:44:44.707

1

It's not working: Let $g(x) = x^2 +1$, $h(x) = x^3 +1$ defined on $[0,1]$. Then $g$ is maximized at $1$, $h$ minimized at $0$, $f(0) = f(1) = 1$, but $f(\frac 12)>1$.

Remark: one can let $g, h$ to be $2$ on $[1, \infty)$ and $1$ on $(-\infty, 0]$.

edited Sep 09 '15 at 01:44

answered Sep 09 '15 at 00:17

I meant for all real numbers, but I'll clarify the question. – P Prog uzz Coderam Golflesming Sep 09 '15 at 00:35
@PProguzzCoderamGolflesming : That's the same. Please see the edit. – Sep 09 '15 at 01:45

Find maximum of quotient of functions

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