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Is this statement true

If both functions \(f\) and \(g\) are increasing at the interval \(I\), then \(fg\) also increasing at the interval \(I\).

This statement is true or false?

Note by Kho Yen Hong
3 years, 3 months ago

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\(f(x)\) and \(g(x)\) are increasing; therefore, \(f'(x)>0\) and \(g'(x)>0\) for all \(x\in I.\) The slope of \(f(x)g(x),\) denoted \((f(x)g(x))',\) is equal to \(f'(x)g(x)+g'(x)f(x).\) A "necessary" condition, as @Calvin Lin says, would be that at least one of \(f(x)\) and \(g(x)\) be positive for all \(x\in I\). A "sufficient" condition to say that \(f(x)g(x)\) is always increasing would be to say that \(f(x)>0\) and \(g(x)>0\) for all \(x\in I.\)

The statement is false. A counterexample is when \(f(x)=x\) and \(g(x)=x+1.\) Then \(f(x)g(x)=x^2+x\) and \((f(x)g(x))'=2x+1.\) Over the range \(\left(\infty,-\frac{1}{2}\right),\) both \(f(x)\) and \(g(x)\) are increasing, but \(f(x)g(x)\) is decreasing. Trevor B. · 3 years, 3 months ago

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if f and g both are greater than 0 then its true .. for the interval I .. else it needs to be checked .. rightly pointed out by aditi .. where y=x <0 for x<0 and e^x is always positive .. !! xe^x then have a critical point atx=-1 which is its minima .. !! Ramesh Goenka · 3 years, 3 months ago

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@Ramesh Goenka Can you state a sufficient condition for \( fg\) to be increasing?

Can you state a necessary condition for \(fg\) to be increasing? Calvin Lin Staff · 3 years, 3 months ago

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That's false. An example: f(x)=x and g(x)=e^x Both are increasing but there product is a non-monotonic function. Aditi Agarwal · 3 years, 3 months ago

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