Chapter 1.2, Problem 128E

Explain why the graphs of a quadratic function (parabola) p(x) and a linear function l (x) can intersect in at most two points. Suppose that p(a) = l(a) and p(b) = l (b), and that ${\displaystyle {\int }_a^{b}p\left(t\right)dt>}{\displaystyle {\int }_a^b\mathcal{l}\left(t\right)dt}$ . Explain why ${\displaystyle {\int }_c^{d}p\left(t\right)dt>}{\displaystyle {\int }_c^d\mathcal{l}\left(t\right)dt}$ whenever $a\le c\le d\le b$ .