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?

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$f(x)$ 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. - 7 years 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 .. !!

Ramesh Goenka - 7 years ago

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 - 7 years 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.

Aditi Agarwal - 7 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$