Which of the following is always correct

Moderate

A

If $f' (x) > 0 \forall x \in$ domain, then $f (x)$ must be one-one
B

If $f' (x) < 0 \forall x \in$ domain, then $f (x)$ must be one-one
C

If $| f (x) |$ be continuous at $x = a$ , then $f (x)$ is also continuous at $x = a$
D

If $f (x)$ is continuous at $x = a$ , $f (a) = 2$ and $x = a$ is the point of local minima of $f (x)$ , then $[f (x)]$ , where [.] denotes greatest integer function, is also continuous at $x = a$

Solution

In (A) $f' (x) > 0$ for every $x \in D_{f}$
$\Rightarrow f (x)$ is one-one

ex. $f (x) = \tan x$

In (B), $f' (x) < 0 \forall x \in D_{f} \Rightarrow f$ is one-one

ex. $f (x) = \cot x$

In (C), $| f (x) |$ is continuous $\Rightarrow f (x)$ is continuous

ex. $f (x)$ $= {\begin{matrix} 1 : x < 0 \\ - 1 : x \geq 0 \end{matrix}$ is discontinuous at $x = 0$

But $| f (x) | = 1 \forall x \in R$ which is continuous

$∴$ (A), (B), (C) are not correct

In (D), $f (a) = 2$ , $x = a$ is a point of local minimum and $f (x)$ is continuous

$∴ f (a + h) > 2, f (a - h) > 2$

$\Rightarrow [f (a + h)] = [f (a - h)] = 2$

Hence $[f (x)]$ is also continuous at $x = a$

$∴$ (D) is correct.

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