Question

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

Answer 1

$ \mathrm{P}^{\prime}(\mathrm{x})>0 \text { and }(\mathrm{P}(\mathrm{x}))^{2}+4 \leq 4 \mathrm{P}\left(\mathrm{x}^{2}\right)$
put $\mathrm{x}=0 \Rightarrow(\mathrm{P}(0))^{2}-4 \mathrm{P}(0)+4 \leq 0$
$\therefore(\mathrm{P}(0)-2)^{2} \leq 0 \Rightarrow \mathrm{P}(0)=2$
$\|$ ly put $\mathrm{x}=1 \Rightarrow(\mathrm{P}(1)-2)^{2} \leq 0$
$\Rightarrow \mathrm{P}(1)=2$
$\therefore$ 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. ]