[数学 - 导数隐零点] 看起来简单？求过定点 (0, √e) 的直线与 y=e^x 交点横坐标差最小时的斜率
题目
已知曲线 ，直线 与 交于 、 两点，设关于 的函数 表示两交点横坐标之差。若 的最小值为 ，求 的值。
题短事大，拿到此题时你不会想到要做多久……
直觉分析
直线 过定点 ，若与 有两个交点，必然一个在 y 轴左侧，一个在 y 轴右侧。
- 若 ，则 y 轴左侧的交点将无限远。
- 若 ，则 y 轴右侧的交点将无限远。
可见， 时，理应存在一个 使得 A、B 横向差距最小，即 取最小值。
要解决这道题，首先肯定是要联立的！
接下来怎么做，就看每个人的习惯了：
不成立之法：作切线
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想用切线的几何意义做题也是人之常情，但注意这里不是两个交点的距离，而是横坐标的距离。直接对 作切线实在是没什么用处。
法一：直接参变分离（超纲难免，韦达小彩蛋）
我一眼看到这个方程就想要参变分离，毕竟 实在是太容易用 表示了。很明显 ，所以：
那么好说， 和 （水平直线）有两个交点呗！简单求一下图形：
这一项，肯定是大于 （）的（求导出来是 ，比较显然），所以 的分子大于 ，分母也显然大于 。 时，分子小于 0。
所以这个函数是两段严格增函数， 前后由正无穷转负无穷。
那么问题来了：什么时候两个交点的横坐标最小呢？
你有没有觉得，是两个交点处切线斜率相同的时候呢？（这个时候因为横坐标之差就是距离了，可以用切线，有兴趣的读者可以对照图像思考一下）