First slide

Monotonicity

Question

If $f (x) = x + \sin x, g (x) = e^{- x}, u = \sqrt{c + 1} - \sqrt{c},$ $v = \sqrt{c} - \sqrt{c - 1}, (c > 1),$ then

Moderate

A

$fog (u) < fog (v)$
B

$gof (u) < gof (v)$
C

$gof (u) > gof (v)$
D

$fog (u) < < fog (v)$

Solution

$u = \sqrt{c + 1} - \sqrt{c} u = \frac{1}{\sqrt{c + 1} + \sqrt{c}} and v = \frac{1}{\sqrt{c - 1} + \sqrt{c}}$
Clearly, $u < v .$
Also, f is increasing whereas g is decreasing. Thus,
$u < v$
or $f (u) < f (v)$
or $gof (u) > gof (v)$

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