Question 162
Let $P$ be a polynomial with positive real coefficients. Prove that if
$$
P\left(\frac{1}{x}\right) \geq \frac{1}{P(x)}
$$
holds for $x=1$, then it holds for every $x \gt 0$.
$$
P\left(\frac{1}{x}\right) \geq \frac{1}{P(x)}
$$
holds for $x=1$, then it holds for every $x \gt 0$.