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?

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## Comments

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TopNewest\(f(x)\) 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.

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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 .. !!

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

Can you state a necessary condition for \(fg\) to be increasing?

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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.

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