Question

A differentiable function $f\left(x\right)$ is strictly increasing $\forall xϵR$, then -

• A. $f^{\prime }\left(x\right)>0\forall xϵR$
• B. $f^{\prime }\left(x\right)\ge 0\forall xϵR$, provided it vanishes at finite number of points.
• C. $f^{\prime }\left(x\right)\ge 0\forall xϵR$ provided it vanishes at discrete points though the number of these discrete points may not be finite.
• D. $f^{\prime }\left(x\right)\ge 0\forall xϵR$ provided it vanishes at discrete points and the number of these of these discrete points must be infinite.

Answer 1

Answer: (C)

$f^{\prime }\left(x\right)\ge 0\forall xϵR$ provided it vanishes at discrete points though the number of these discrete points may not be finite.

For
$f\left(x\right)$ to be strictly increasing, the slope of $y=f\left(x\right)$ at any real $x$ should be greater than $0.$
That is
$f^{\prime }\left(x\right)>0$
But it vanishes at discrete points though the number of these discrete points may not be finite.
Hence answer is C.