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Given a couple of strictly positive functions, is optimizing the product of these functions equivalent to optimizing their sum? Any proof or counter example in the literature?

Mpampa
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  • Note that if you want to optimize a product of two strictly positive functions, you can take a logarithm in order to obtain an expression that is a sum: $\mathrm{min}[f(x)\cdot g(x)]=\mathrm{min}[\ln f(x)+\ln g(x)]$. – Karlo Aug 31 '17 at 09:47

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No. Counterexample: take $f(x)=(x-1)^2+1$ and $g(x)=2x^2 + 1$. Then (calculated quickly using WolframAlpha): $$\mathrm{min}[f(x)+g(x)]=\frac 1 3$$ $$\mathrm{min}[f(x)\cdot g(x)]\approx 0.273301$$

Karlo
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    That was quick and straight to the point. Thanks Karlo. – Mpampa Aug 31 '17 at 09:53
  • I am wondering what condition must hold in order for those two to have the same extrema. – Mpampa Aug 31 '17 at 09:54
  • An example where your statement is satisfied: $f(x)=x^2 + 1,g(x)=(x-1)^2 + 1$, for which both the sum and the product are minimal for $x=\frac 1 2$. – Karlo Aug 31 '17 at 10:25