Question

Let $f:R\to R$ be a differentiable function for all values of $x$ and has the peoperty that $f\left(x\right)$ and $f^{\prime }\left(x\right)$ have opposite signs for all values of $x$. Then,

• A. $f\left(x\right)$ is an increasing function
• B. $f\left(x\right)$ is a decreasing function
• C. $f^2\left(x\right)$ is a decreasing function
• D. $|f\left(x\right)|$ is an increasing function

Answer 1

The correct option is C $f^2\left(x\right)$ is a decreasing function

Given $f\left(x\right)$ and $f^{{}^{\prime }}\left(x\right)$ have opposite signs

$\Rightarrow f\left(x\right)f^{{}^{\prime }}\left(x\right)<0$

We cannot say f(x) is increasing or decreasing because we don't know the sign of f(x)

Let us take option C;

If $f^2\left(x\right)$ is increasing $\Rightarrow \frac{d}{dx}\left(f^2\right(x\left)\right)>0$

$\Rightarrow 2f\left(x\right)f^{{}^{\prime }}\left(x\right)>0$

$\Rightarrow f\left(x\right)f^{{}^{\prime }}\left(x\right)>0$ which is the given condition

$\therefore$option C is correct