= Solution
Choose a sufficiently large positive $n$ for which $g_n=B_n^{-1}f>0$ throughout the interval. Expanding $B_n(g_n)=f$ gives
$$
f(x)=\sum_{k=0}^n\binom nk g_n(k/n)x^k(1-x)^{n-k}.
$$
Therefore the coefficients
$$
\boxed{c_k=\binom nk g_n(k/n)>0,\qquad 0\le k\le n,}
$$
supply 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.
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