$f\left(x\right)=x\left(4x-3\right)\left(x-2\right)$，且 $f^{\prime}\left(x\right)=12x^2-22x+6=2\left(3x-1\right)\left(2x-3\right)$。$f^{\prime}\left(1\right)=-4<0$，故 $(1)$ 錯。相對極大值 $f\left(\dfrac{1}{3}\right)=\dfrac{25}{27}<1$，故與 $y=1$ 僅交一點，$(2)$ 錯。$f\left(\dfrac{3}{2}\right)=-\dfrac{9}{4}$，為唯一相對極小值。又 $\pi>2$，故 $f\left(\pi\right)>0$。而 $\cos\dfrac{4\pi}{7}<0$，故 $f\left(\cos\dfrac{4\pi}{7}\right)<0$。