# 2017年考研数二第18题解析：导数、函数极值、单调性

## 解析

### 一、计算 $\frac{\mathrm{d} y}{\mathrm{d} x} = 0$.

$$3x^{2} + 3y^{2} \frac{\mathrm{d} y}{\mathrm{d} x} – 3 + 3 \frac{\mathrm{d} y}{\mathrm{d} x} = 0 \Rightarrow$$

$$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{3 – 3x^{2}}{3 + 3y^{2}}.$$

$$3 – 3 x^{2} = 0 \Rightarrow$$

$$x = -1 或 x = 1.$$

### 二、当 $x \in (- \infty, -1)$ 时，$\frac{\mathrm{d} y}{\mathrm{d} x} < 0$.

$$\lim_{x \rightarrow – \infty} (x^{3} + y^{3} – 3x + 3y – 2 = 0) \approx$$

$$\lim_{x \rightarrow – \infty} (x^{3} + y^{3} = 0).$$

$$\lim_{x \rightarrow – \infty} (3x^{2} + 3y^{2} \frac{\mathrm{d} y}{\mathrm{d} x} – 3 + 3 \frac{\mathrm{d} y}{\mathrm{d} x} = 0) \approx$$

$$\lim_{x \rightarrow – \infty} (3x^{2} + 3y^{2} \frac{\mathrm{d} y}{\mathrm{d} x} = 0)$$

$$(\lim_{x \rightarrow – \infty} x^{3}) \Rightarrow – \infty \Rightarrow$$

$$(\lim_{x \rightarrow – \infty} x) \Rightarrow – \infty.$$

$$(\lim_{x \rightarrow – \infty} y^{3}) \Rightarrow + \infty \Rightarrow$$

$$(\lim_{x \rightarrow – \infty} y) \Rightarrow + \infty.$$

$$(\lim_{x \rightarrow – \infty} x^{2}) \Rightarrow + \infty;$$

$$(\lim_{x \rightarrow – \infty} y^{2}) \Rightarrow + \infty;$$

$$\frac{\mathrm{d} y}{\mathrm{d} x} < 0, x \in (- \infty ,-1).$$

### 三、当 $x \in (1, – \infty)$ 时，$\frac{\mathrm{d} y}{\mathrm{d} x} < 0$.

$$\lim_{x \rightarrow + \infty} (x^{3} + y^{3} – 3x + 3y – 2 = 0) \approx$$

$$\lim_{x \rightarrow + \infty} (x^{3} + y^{3} = 0).$$

$$\lim_{x \rightarrow + \infty} (3x^{2} + 3y^{2} \frac{\mathrm{d} y}{\mathrm{d} x} – 3 + 3 \frac{\mathrm{d} y}{\mathrm{d} x} = 0) \approx$$

$$\lim_{x \rightarrow + \infty} (3x^{2} + 3y^{2} \frac{\mathrm{d} y}{\mathrm{d} x} = 0)$$

$$(\lim_{x \rightarrow + \infty} x^{3}) \Rightarrow + \infty \Rightarrow$$

$$(\lim_{x \rightarrow + \infty} x) \Rightarrow + \infty.$$

$$(\lim_{x \rightarrow + \infty} y^{3}) \Rightarrow – \infty.$$

$$(\lim_{x \rightarrow + \infty} y) \Rightarrow – \infty.$$

$$(\lim_{x \rightarrow + \infty} x^{2}) \Rightarrow + \infty;$$

$$(\lim_{x \rightarrow + \infty} y^{2}) \Rightarrow + \infty;$$

$$\frac{\mathrm{d} y}{\mathrm{d} x} < 0, x \in (1 , + \infty).$$

### 三、当 $x \in (- 1, 1)$ 时，$\frac{\mathrm{d} y}{\mathrm{d} x} > 0$.

$$0 < y < 1.$$

$$3x^{2} + 3y^{2} \frac{\mathrm{d} y}{\mathrm{d} x} – 3 + 3 \frac{\mathrm{d} y}{\mathrm{d} x} = 0 \Rightarrow$$

$$3y^{2} \frac{\mathrm{d} y}{\mathrm{d} x} – 3 + 3 \frac{\mathrm{d} y}{\mathrm{d} x} = 0 \Rightarrow$$

$$(3y^{2} + 3) \frac{\mathrm{d} y}{\mathrm{d} x} = 3 \Rightarrow$$

$$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{3}{3y^{2} + 3}.$$

$$\frac{3}{3y^{2} + 3} > 0 \Rightarrow$$

$$\frac{\mathrm{d} y}{\mathrm{d} x} > 0, x \in (-1,1).$$

### 四、综上可知

$$\left\{\begin{matrix} \frac{\mathrm{d} y}{\mathrm{d} x} < 0, x \in (- \infty ,-1);\\ \frac{\mathrm{d} y}{\mathrm{d} x} > 0, x \in (-1,1);\\ \frac{\mathrm{d} y}{\mathrm{d} x} < 0, x \in (1 , + \infty). \end{matrix}\right.$$

1. 当 $x = -1$ 时，函数 $y(x)$ 取得极小值，且极小值为 $y(-1) = 0$;
2. 当 $x = 1$ 时，函数 $y(x)$ 取得极大值，且极大值为 $y(1) = 1$.