曲线 $C$ 的方程为 $y=\dfrac{5x^2}{5x-2}$。

(a) 求 $C$ 的渐近线方程。

(b) 求 $C$ 上驻点的坐标。

(c) 画出 $C$ 的草图。

(d) 画出方程为 $y=\Big|\dfrac{5x^2}{5x-2}\Big|$ 的曲线的草图，并用精确形式求出满足 $\Big|\dfrac{5x^2}{5x-2}\Big|<2$ 的 $x$ 的取值集合。

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。

多项式除法得到斜渐近线，分母零点不能约消。

Hence the asymptotes are $\boxed{x=\frac25}$ and $\boxed{y=x+\frac25}$.

用商法则并因式分解分子，两个驻点都在定义域内。

Setting the numerator to zero gives $x=0$ or $x=4/5$. Substitution yields

The derivative changes from positive to negative at $x=0$ and from negative to positive at $x=4/5$, giving a maximum and minimum respectively.

左支非正并在原点平滑接触横轴，右支有正的最低点。余项符号决定两支与斜渐近线的相对位置。

Label $x=2/5$ and $y=x+2/5$. For $x<2/5$, sketch a branch below the oblique asymptote, rising from $-\infty$ to touch the $x$-axis at its maximum $(0,0)$ and then falling to $-\infty$ as $x\to(2/5)^-$.

For $x>2/5$, sketch a branch above the oblique asymptote, falling from $+\infty$ to the minimum $(4/5,8/5)$ and then rising to $+\infty$.

Both outer tails approach the oblique asymptote, and $(0,0)$ is the only axis intersection.

反射整条左支，原点仍为光滑最低点。解绝对值不等式时分别在竖直渐近线两侧处理分母符号。

For the absolute-value graph, reflect the branch with $x<2/5$ in the $x$-axis, retaining the branch with $x>2/5$. The origin becomes a smooth minimum, the right minimum remains $(4/5,8/5)$, and both sides tend to $+\infty$ at $x=2/5$.

The outer asymptotes are $y=-x-2/5$ on the left and $y=x+2/5$ on the right.

For $x<2/5$, the original function is nonpositive, so the inequality is equivalent to $5x^2/(5x-2)>-2$. Multiplication by the negative denominator reverses the sign:

The roots are $-1\pm3\sqrt5/5$, both less than $2/5$. Hence this gives

For $x>2/5$, the original function is positive, and multiplication by the positive denominator gives

Its roots are $1\pm\sqrt5/5$, both greater than $2/5$. Combining the two cases,