Question

Let $f$ be a continuous and differentiable function such that $f\left(x\right)$ and $f^{\prime }\left(x\right)$ have opposite signs everywhere. Then

• A. $f$ is increasing
• B. $f$ is decreasing
• C. $|f|$ is increasing and decreasing
• D. $|f|$ is decreasing.

Answer 1

The correct option is D $|f|$ is decreasing.

According to Darboux Theorem

If $f\left(x\right)$ is differential in the closed interval $[a,b]$ and $f^1\left(a\right)$ and $f^1\left(b\right)$ are of opposite signs,then there is a point $C\in (a,b)i.ea<c<b$ such that $f^1\left(c\right)=0$ and hence the function is decreasing

Therefore option D is the correct answer