= Solution
The <Markov interlacing lemma> says that <differentiation> preserves <interlacing roots of polynomials>. More explicitly, if two degree-$d$ real <polynomials> have ordered real roots satisfying $a_1\le b_1\le a_2\le\cdots\le a_d\le b_d$, their $k$th <derivatives> have ordered roots $a_j^{(k)},b_j^{(k)}$ satisfying
$$
a_1^{(k)}\le b_1^{(k)}\le a_2^{(k)}\le\cdots\le a_{d-k}^{(k)}\le b_{d-k}^{(k)},\qquad 0\le k<d.
$$
The analogous alternating order is preserved for <polynomials> of consecutive degrees. Common roots are allowed through weak inequalities; strictly interlacing simple roots give strict inequalities. When a derivative is constant, its root list is empty.
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