Upper bound of a uniformly converging sequence of polynomials

Question

Let $k\geq 2$, and let $P_k$ be a sequence of polynomials, such that:

1. $P_k=\sum_{n=2}^{k+1}a_{n,k}X^n \in \mathbb{Q}[X]$, $a_{2,k}\neq 0$, $\deg P_k \leq k+1$, and consider $P_k :[0,1]\rightarrow \mathbb{R}$ as a real valued function.
2. $P_k(1)=\frac{1}{k(k+1)}$ and $\mid a_{n,k}\mid < \frac{2}{k}$, for all $n$ and $k$.
3. for any $a<1$, $\max_{x\in[0,a]} \mid P_k(x)\mid \rightarrow 0$, as $k\rightarrow +\infty$.

Questions Assume we are given a sequence of polynomials $(P_k)_{k\geq 2}$ satisfying conditions 1,2,3 from above. Do we have $\max_{x\in[0,1]} \mid P_k(x)\mid =O(\frac{1}{k^2})$, for $k$ large enough?

I am sorry if this question is trivial, however analysis is not my field. This question is related to a question that I asked before.

Answer 1

Take $$ P_k(x)=\frac{x^2}{k(k+1)}+\frac{x^2(1-x)}k. $$