曲线 $C$ 的方程为 $y=\dfrac{x+2}{x^2+3x+1}$。

(a) 求 $C$ 的渐近线方程。

(b) 证明 $C$ 没有驻点。

(c) 画出 $C$ 的草图，并写出曲线与坐标轴交点的坐标。

(d) 画出方程为 $y=\Big|\dfrac{x+2}{x^2+3x+1}\Big|$ 的曲线的草图。

(e) 求满足 $\Big|\dfrac{x+2}{x^2+3x+1}\Big|>2$ 的 $x$ 的值的集合，以精确形式表示。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

分母两个实根都不能约去，给出两条竖直渐近线；分母次数较高，水平渐近线为零。

Solving $x^2+3x+1=0$ gives

The numerator is non-zero at both roots. Thus the vertical asymptotes are

Since the denominator has higher degree than the numerator, the horizontal asymptote is $\boxed{y=0}$.

商法则求导，配方说明导数分子恒负，因此整个定义域内均不可能有驻点。

By the quotient rule,

The derivative is never zero wherever $C$ is defined. Hence $\boxed{C\text{ has no stationary points}}$.

标明两渐近线和两轴交点，每一支都严格递减。中间一支要穿过横轴。

Write $a=(-3-\sqrt5)/2$ and $b=(-3+\sqrt5)/2$. The intercepts are $\boxed{(-2,0)}$ and $\boxed{(0,2)}$.

Sketch three decreasing branches with vertical asymptotes $x=a,x=b$ and horizontal asymptote $y=0$:

• For $x<a$, the branch approaches $0$ from below as $x\to-\infty$ and tends to $-\infty$ as $x\to a^-$.
• For $a<x<b$, it decreases from $+\infty$ to $-\infty$, crossing the axis at $(-2,0)$.
• For $x>b$, it decreases from $+\infty$, passes through $(0,2)$ and approaches $0$ from above.

绝对值只反射负函数值的部分，保留两条竖直渐近线，并在原横轴交点形成尖点。

Reflect the portions with $x<a$ and $-2<x<b$ in the $x$-axis. Leave the positive portions unchanged.

The transformed graph lies on or above the $x$-axis, has a sharp minimum at $(-2,0)$, retains the intercept $(0,2)$ and both vertical asymptotes, and approaches $y=0$ from above at both ends.

两侧非负，所以可以平方，再乘正的分母平方。解出乘积的符号区间后，必须排除两条竖直渐近线。

Let $D=x^2+3x+1\ne0$. Squaring the inequality and multiplying by $D^2>0$ gives

Factorising,

The four distinct roots in increasing order are

The product has positive leading coefficient, so its sign alternates at these simple roots. It is negative on $(c,-5/2)$ and $(d,0)$.

Excluding the undefined points from (a), the required set is

where $c=(-7-\sqrt{17})/4$ and $d=(-7+\sqrt{17})/4$.