Question

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

• A. $f^{'} \left(x\right)$ 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)

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