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For $f$ of degree $d$ and $n\ge \max (1,d)$, compute its degree-$n$ Bernstein basis coefficients and solve $\max \lambda$ subject to $\lambda \le {\beta }_{k,n}$, $0\le k\le n$. This is a linear program with $n+1$ inequalities. The coefficients form a convex-combination enclosure of the polynomial values, so $v_n\le \min f$. Degree elevation of Bernstein coefficients makes the bounds monotone. For every $ϵ>0$, the polynomial $f-(\min f-ϵ)$ is strictly positive, so positive Bernstein coefficients for a strictly positive polynomial gives an eventual feasible bound $\min f-ϵ$. Hence $v_n$ converges to the true minimum. Earlier indices $n<d$ can use a one-inequality program at any common coefficient-based lower bound, preserving the stated constraint count for every positive index.

First address the positivity request in part (a), which has no separate Solution heading in the stub. For $x\in [0,1]$, each $x^k(1-x{)}^{n-k}$ and each coefficient $c_k$ is nonnegative. Their sum is therefore nonnegative, including at the endpoints:This is the elementary direction of the positive Bernstein coefficients for a strictly positive polynomial criterion; the converse developed later needs strict positivity.

For completeness, the convergence fact listed as (i) also has a short proof. If $K\sim \mathrm{Bin}(n,x)$, the Bernstein polynomial is $B_n(f)(x)=\mathbb{E}[f(K/n)]$. Its mean argument is $x$ and $\mathrm{Var}(K/n)=x(1-x)/n\le 1/(4n)$. Given $ϵ>0$, uniform continuity of $f$ on the compact interval supplies $\delta >0$ such that $|f(s)-f(t)|<ϵ$ whenever $|s-t|\le \delta$. By Chebyshev's inequality, uniformly in $x$,Letting $n\to \infty$ and then $ϵ\downarrow 0$ proves $\boxed{\Vert B_n(f)-f{\Vert }_{\infty }\to 0}$. The two numbered items in the source are assumptions supplied for later parts, not additional exam questions; their stub sections contain these supporting derivations.

Choose a sufficiently large positive $n$ for which $g_n=B_n^{-1}f>0$ throughout the interval. Expanding $B_n(g_n)=f$ givesTherefore the coefficientssupply the requested representation, in particular with nonnegative coefficients. This proves positive Bernstein coefficients for a strictly positive polynomial. Strict positivity cannot generally be weakened to mere nonnegativity: a nonzero polynomial such as $(x-1/2{)}^2$ vanishes at an interior point, whereas every basis term is positive there, so no nonnegative-coefficient representation could vanish unless every coefficient were zero.