Question 4
Let $\alpha>1$ be real, and let $P(x)$ be a real polynomial of degree $24$ whose leading and constant coefficients are both $1$. Suppose $P(x)$ has $24$ positive real roots, all at most $\alpha$. Prove that
$$|P(1)|\le\left(\frac{19}{5}\right)^5(\alpha-1)^{24}.$$
$$|P(1)|\le\left(\frac{19}{5}\right)^5(\alpha-1)^{24}.$$