Question

Let a continuous and differentiable function $f\left(x\right)$ is such that $f\left(x\right)$ and $\frac{d}{dx}f\left(x\right)$ have opposite signs everywhere. Then,

• A. $f^{{}^{\prime }}\left(x\right)$ is always increasing
• B. $f\left(x\right)$ is always increasing
• C. $\left|f\left(x\right)\right|$ is non-decreasing
• D. $\left|f\left(x\right)\right|$ is decreasing

Answer 1

Answer: (D)

$|f(x)|=\{\begin{array}{lll}f(x)& :& f(x)\ge 0\\ -f(x)& :& f(x)<0\end{array}$
$\frac{d}{dx}(|f(x)|)=\{\begin{array}{lll}{f}^{\prime }(x)& :& f(x)\ge 0\\ -f^{\prime }(x)& :& f(x)<0\end{array}$
As $f(x)$ and $f^{\prime }(x)$ have opposite signs, then $\frac{d}{dx}(|f(x)|)$ is negative everywhere, hence $|f(x)|$ is a decreasing function.