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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, 10 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.

- 3 years, 9 months ago

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 .. !!

- 3 years, 9 months ago

Can you state a sufficient condition for $$fg$$ to be increasing?

Can you state a necessary condition for $$fg$$ to be increasing?

Staff - 3 years, 9 months ago

That's false. An example: f(x)=x and g(x)=e^x Both are increasing but there product is a non-monotonic function.

- 3 years, 10 months ago