Given two real polynomials $f,g$ with the roots are all real. We say the zeros of $f,g$ interlace if $$\alpha_1\leq\beta_1\leq\alpha_2\leq\beta_2\leq\ldots$$ or $$\beta_1\leq\alpha_2\leq\beta_2\leq\alpha_2\leq\ldots$$ where $\alpha_i,\beta_j$ are the roots of $f,g$, respectively. Define the Wroskian $W[f,g]=f^{'}g-fg^{'}$.
How to prove the following result?
If the zeros of $f,g$ interlace, then $W[f,g]$ is either nonnegative or nonpositive on the whole real axis $\mathbb{R}$.
I'll give you a hint. Try to prove the following.
Note that we necessarily have the roots of $f,g$ are distinct, and $\lvert\deg f-\deg g\rvert\leq 1$. Furthermore we can write (assuming WLOG $\deg f\leq\deg g$) $f=ag+\sum_{j=1}^{\deg g}b_jg_j$ where $g_j(x):=g(x)/(x-\beta_j)$, unique $(a,b_1,\dots,b_{\deg g})\in\mathbb{R}^{1+\deg g}$ since the $\beta_j$ are distinct. The interlacing condition is equivalent to: $b_j$, for all $j$, are of the same sign. So $$ \frac{W[f,g]}{g^2}=\left(\frac{f}g\right)'=\sum_j\frac{-b_j}{(x-\beta_j)^2} $$ or equivalently, $W[f,g]=-\sum_j b_jg_j^2$.