Question 5
Let $a_0,a_1,\ldots,a_n,a_{n+1}$ be a finite sequence of real numbers satisfying $a_0=a_{n+1}=0$ and
$$|a_{k-1}-2a_k+a_{k+1}|\le1.$$
Prove that
$$a_k\le\frac{k(n+1-k)}2\qquad(k=0,1,\ldots,n+1).$$
$$|a_{k-1}-2a_k+a_{k+1}|\le1.$$
Prove that
$$a_k\le\frac{k(n+1-k)}2\qquad(k=0,1,\ldots,n+1).$$