Question 2
Let
$$
f(x) = a_{0} + a_{1} x + a_{2} x^{2} + \cdots + a_{n} x^{n}
$$
be a polynomial with coefficients satisfying the conditions:
$$
0 \leq a_{i} \leq a_{0}, \quad i = 1, 2, \ldots, n
$$
Let $b_{0}, b_{1}, \ldots, b_{2n}$ be the coefficients of the polynomial
$$
\begin{aligned}
(f(x))^{2} &= \left(a_{0} + a_{1} x + a_{2} x^{2} + \cdots + a_{n} x^{n}\right)^{2} \\
&= b_{0} + b_{1} x + b_{2} x^{2} + \cdots + b_{n+1} x^{n+1} + \cdots + b_{2n} x^{2n}
\end{aligned}
$$
Prove that
$$
b_{n+1} \leq \frac{1}{2} (f(1))^{2}
$$
$$
f(x) = a_{0} + a_{1} x + a_{2} x^{2} + \cdots + a_{n} x^{n}
$$
be a polynomial with coefficients satisfying the conditions:
$$
0 \leq a_{i} \leq a_{0}, \quad i = 1, 2, \ldots, n
$$
Let $b_{0}, b_{1}, \ldots, b_{2n}$ be the coefficients of the polynomial
$$
\begin{aligned}
(f(x))^{2} &= \left(a_{0} + a_{1} x + a_{2} x^{2} + \cdots + a_{n} x^{n}\right)^{2} \\
&= b_{0} + b_{1} x + b_{2} x^{2} + \cdots + b_{n+1} x^{n+1} + \cdots + b_{2n} x^{2n}
\end{aligned}
$$
Prove that
$$
b_{n+1} \leq \frac{1}{2} (f(1))^{2}
$$