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  4. If f ( x )=3 tan x +(3 a +1) ln | sec x |+( a -3) x is increasing in (0, (π/2)) then minimum integral value of

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Q. If $f ( x )=3 \tan x +(3 a +1) \ln |\sec x |+( a -3) x$ is increasing in $\left(0, \frac{\pi}{2}\right)$ then minimum integral value of

Application of Derivatives

Solution:

$f^{\prime}(x)=3 \sec ^2 x+(3 a+1) \tan x+(a-3) $
$=3 \tan ^2 x+3 a \tan x+\tan x+a $
$=(3 \tan x+1)(\tan x+a)$
$f ^{\prime}( x )>0 \forall x \in\left(0, \frac{\pi}{2}\right) $
$\Rightarrow \tan x+a>0 \forall x \in\left(0, \frac{\pi}{2}\right)$
$\therefore a \geq 0$
$\therefore \text { minimum integral value of } a=0 $