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Suppose, I need to maximize the function $\frac{a(t)}{b(t)}$ over some continuous domain of $t$, and I know the maximum exists.

WLOG, the maximum is obtained at $t_1$.

Now I'm wondering whether if $\frac{a(t)+C_1}{b(t) + C_2}$, where $C_1, C_2 > 0$, is also maximized at $t_1$?

coudy
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peng yu
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1 Answers1

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Counterexample:

Take $a(t) = t$ and $b(t) = -k$ where $k>0$, over the domain $[t_0,t_1]$ where $t_0<0<t_1<|t_0|$

The maximum of $a(t) \over b(t)$ occurs at $t=t_0$. However, if $C_2>k, \forall t\geq0$; \begin{equation} \frac{a(t_0)}{b(t_0)}<0<\frac{a(t)}{b(t)} \end{equation}

by24
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