二次函数 $y=a x^{2}-4 a x+c(a>0)$ 的图象过 $A\left(-2, y_{1}\right), B\left(0, y_{2}\right), C\left(3, y_{3}\right), D\left(5, y_{4}\right)$ 四个点, 下列说法中一定正确的是( )

 $\because y=a x^{2}-4 a x+c, \therefore$ 拋物线的对称轴为直线 $x=-\frac{-4 a}{2 a}=2, \because a>0,: \therefore$ 拋物线开口向上, $\because 2-(-2)>5-2>2-0>3-2, \quad \therefore y_{1}>y_{4}>y_{2}>y_{3}$,若 $y_{1}>y_{4}>y_{2}>0>y_{3}$, 则 $y_{1} y_{2}>0, y_{3} y_{4}<0$, 选项 $A$ 错误.

若 $y_{1}>y_{4}>y_{2}>0>y_{3}$, 则 $y_{1} y_{4}>0, y_{2} y_{3}<0$, 选项 $B$ 错误.

若 $y_{2} y_{4}<0$, 则 $y_{1}>y_{4}>0>y_{2}>y_{3}, \therefore y_{1} y_{3}<0$, 选项 $C$ 正确.

若 $y_{1}>y_{4}>y_{2}>0>y_{3}$, 则 $y_{3} y_{4}<0, y_{1} y_{2}>0$, 选项 $D$ 错误.

