I have two functions $f(x)>0$ and $g(x)>0$, both decreasing. Then, can I claim that the product $f(x)g(x)$ is decreasing as well? ($x>0$)
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1That is obvious, by the very definition of‘decreasing’. – Bernard Aug 30 '20 at 23:28
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Yes since they are positive functions – alphaomega Aug 30 '20 at 23:31
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Quick beginner guide for asking a well-received question + please avoid "no clue" questions. – Anne Bauval May 23 '23 at 05:22
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Yes, $0<a<b$ and $0<c<d$ implies $ac <bc <bd$ so $ac <bd$.
Let $x<y$ and take $a=f(x), b=f(y), c=g(x), d=f(y)$.
Kavi Rama Murthy
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Thanks. Does this claim work for two increasing functions as well? I mean, is the product of two increasing (and positive) functions, increasing? – katy98 Aug 30 '20 at 23:32
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Assume $y>x$. We have $f(x)\geq f(y)$, and since $g(x)>0$ it follows that $f(x)g(x)\geq f(y)g(x)$. Also, we have $g(x)\geq g(y)$ and since $f(y)>0$ we get $g(x)f(y)\geq g(y)f(y)$. Combining this together we indeed get $f(x)g(x)\geq f(y)g(y)$.
Mark
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