First slide

Properties of ITF

Question

Let f(x)=sin x +cos x+ tan x+ arc sin x+ arc cos x + arc tan x . If M and m are maximum and minimum values of f(x) , then their arithmetic mean is equal to

Difficult

A

$\frac{π}{2} + \cos 1$
B

$\frac{π}{2} + \sin 1$
C

$\frac{π}{4} + \tan 1 + \cot 1$
D

None of these

Solution

Domain of f is [-1,1]
$f (x) = sinx + cosx + tanx + \sin^{- 1} x + \cos^{- 1} x + \tan^{- 1} x$
$f (x) = sinx + cosx + tanx + \frac{π}{2} + \tan^{- 1} x$
$f' (x) = cosx - sinx + \sec^{2} x + 0 + \frac{1}{1 + x^{2}}$
here, $\sec^{2} \geq 1$ and $\frac{1}{1 + x^{2}} \in [\frac{1}{2}, 1]$ and $cosx - sinx \in [- \sqrt{2}, \sqrt{2}]$
hence $f' (x) > 0 \Rightarrow$ f is increasing
Range is [f(-1),f(1)]
$∴ {[f (x)]}_{\min} = f (- 1) = - \sin 1 + \cos 1 - \tan 1 + \frac{π}{2} - \frac{π}{4}$
$m = \frac{π}{4} + \cos 1 - \sin 1 - \tan 1$
$∴ {[f (x)]}_{m a x} = f (1) = \sin 1 + \cos 1 + \tan 1 + \frac{π}{2} + \frac{π}{4}$
$M = \frac{3 π}{4} + \cos 1 + \sin 1 + \tan 1$
$\Rightarrow \frac{M + m}{2} = \frac{π}{2} + \cos 1$

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