Choosing right options based on given condition of differentiabile function

Question

Let $f:R\rightarrow R$ be a differentiable function such that $f(x)f'(x)<0$ for all $x$. My question is based on this information, we have to tell whether the $|f(x)|$ is a decreasing function or $f(x)$.

My approach

Since, we have been given the following condition:

$f(x)f'(x)<0$

Taking integral both sides:

$\int f(x)f'(x)dx<0$

Which gives:

$\frac{(f(x))^2}{2}<0$. Hence, we can take square root both sides which gives:

$|f(x)|<0$.

Hence, we can conclude that $|f(x)|$ is a decreasing function. Are my steps correct? I am not sure. Please look into it. Any suggestions would be very beneficial for me. Thanks

Answer 1

If $\phi(x)= {1 \over 2} f^2(x)$ then $\phi'(x) = f(x) f'(x) < 0$.

Hence $\phi$ (and hence $|f|$) is a decreasing function.