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Q.
The least integral value of $f(x)=\frac{(x-1)^7+3(x-1)^6+(x-1)^5+1}{(x-1)^5} \forall x>1$, is equal to
Application of Derivatives
Solution:
Let $x -1= t >0$
$\therefore f ( x )=\frac{ t ^7+3 t ^6+ t ^5+1}{ t ^5}= t ^2+3 t +1+\frac{1}{ t ^5} \\
\Theta \text { A.M. } \geq GM . $
$\left.\Rightarrow \frac{ t ^2+ t + t + t +1+\frac{1}{ t ^5}}{6} \geq \sqrt[6]{ t ^2 \cdot t \cdot t \cdot t \cdot 1 \cdot \frac{1}{ t ^5}} \right]$
$\Rightarrow f ( x ) \geq 6$
$\therefore \text { least integral value }=6 $