Question

Consider the following statements:
Statement I:
$x>\sin x$ for all $x>0$
Statement II:
$f\left(x\right)=x-\sin x$ is an increasing function for all $x>0$
Which one of the following is correct in respect of the above statements?

• A. Both Statements I and II are true and Statement II is the correct explanation of Statement I
• B. Both Statements I and II are true and Statement II is the not correct explanation of Statement I
• C. Statement I is true but Statement II is false
• D. Statement I is true but Statement II is true

Answer 1

The correct option is A Both Statements I and II are true and Statement II is the correct explanation of Statement I

From the graph drawn above, we can see that $y=f\left(x\right)$ is always greater than $y=sinx$ for all $x>0$ i.e. $x>sinx$ for $x>0$

$f\left(x\right)=x-sinx$

$f^{\prime }\left(x\right)=1-cosx$

Now, $cosx\in [-1,1]\therefore (1-cosx)\in [0,2]$

Thus, $f^{\prime }\left(x\right)⩾0$ for all $x$

$f\left(x\right)$ is an increasing function for all $x>0$