一条曲线满足 $\frac{dy}{dx} = 3x^2 + 10x - 8$。

(a) 求使得当 $x$ 增大时 $y$ 减小的 $x$ 的取值范围。

(b) 已知该曲线的极大点的 $y$ 坐标为 27。求曲线方程。

曲线下降时一阶导数为负。先找导数的零点，再判断二次式的符号。

The derivative is $(3x-2)(x+4)$. Its zeros are $x=-4$ and $x=\frac23$. As its leading coefficient is positive, it is negative between the two zeros. Thus the curve decreases for $-4<x<\frac23$.

根据 (a) 的导数符号找极大点的横坐标。积分后用该点的纵坐标求常数。

The derivative changes from positive to negative at $x=-4$, so the maximum point is $(-4,27)$. Integrating,

Substituting the point gives $27=-64+80+32+C=48+C$, so $C=-21$. The curve is $y=x^3+5x^2-8x-21$.