Waste less time on Facebook — follow Brilliant.

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, 1 month ago

No vote yet
1 vote


Sort by:

Top Newest

\(f(x)\) 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, 1 month ago

Log in to reply

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, 1 month ago

Log in to reply

@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, 1 month ago

Log in to reply

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, 1 month ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...