I am working on an exercise on ODE and I need the following lemma to solve it. *Lemma:* Let two functions $y_1, y_2$: $(x_0, x_1)tomathbb{R}$, such that $y_1(x_0) = y_2(x_0)$ and $y_1(x_1) = y_2(x_1)$. Moreover $y_1(x)>y_2(x) ; forall xin (x_0, x_1)$. Prove that there exists some point $zin(x_0,x_1)$, for which $y_1′(z) > y_2′(z)$.

It is geometrically obvious, as every point near enough $x_0$ satisfies the condition, but I couldn’t prove it formally. I tried working with local maximums and minimums, but I think there is a more obvious way than considering all options for $max$ and $min$.