Question

Find the number of polynomials $P(x)$ with integer coefficients such that $P^{\prime }(x)>0$ and $(P(x){)}^2+4\le 4P\left(x^2\right)$ for all $x$

Answer 1

Answer: 0000

${\mathrm{P}}^{\prime }(\mathrm{x})>0\text{ and }(\mathrm{P}(\mathrm{x}){)}^2+4\le 4\mathrm{P}\left({\mathrm{x}}^2\right)$
put $\mathrm{x}=0\Rightarrow (\mathrm{P}(0){)}^2-4\mathrm{P}(0)+4\le 0$
$\therefore (\mathrm{P}(0)-2{)}^2\le 0\Rightarrow \mathrm{P}(0)=2$
$\Vert$ ly put $\mathrm{x}=1\Rightarrow (\mathrm{P}(1)-2{)}^2\le 0$
$\Rightarrow \mathrm{P}(1)=2$
$\therefore$ using Rolle's theorem in $[0,1]{\mathrm{P}}^{\prime }(\mathrm{c})=0$ for some $\mathrm{c}\in (0,1)$ but given ${\mathrm{P}}^{\prime }(\mathrm{x})>0$. Hence no polynomials exists. ]