Question

Find the minimum value of $\frac{(5+x)(2+x)}{(1+x)}$.

Answer 1

$f\left(x\right)=\frac{(5+x)(2+x)}{(1+x)}=\frac{x^2+7x+10}{1+x}$

$f^1\left(x\right)=\frac{x^2+2x-3}{(1+x{)}^2}=0$

$\Rightarrow x=1,-3$

$f^{11}\left(x\right)=\frac{8(x+1)}{(1+x{)}^4}$

$f^{11}\left(1\right)>0,f^{11}(-3)<0$

Hence the minimum value of $f\left(x\right)$ occurs at $x=1$

$\Rightarrow f\left(1\right)=9$

The minimum value of $f\left(x\right)$ is $9$