求曲线 $y=\mathrm{e}^{2x}\sin2x$ 在 $0\le x\le\frac12\pi$ 范围内的驻点的精确坐标。

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。

用乘积法则求导。指数函数恒正，因此只需解三角函数因子的零点；最后把横坐标代回原式求纵坐标。

Differentiating,

At a stationary point, $e^{2x}>0$, so

Here $\cos2x=0$ would not satisfy this equation. Hence

Since $0\le2x\le\pi$, the only solution is $2x=3\pi/4$. Thus

The stationary point is

求导后的三角方程也可以用倍角公式改写为关于 $\tan x$ 的二次方程。注意端点 $x=\pi/2$ 不能在除以 $\cos^2x$ 时遗漏检查。

The derivative is

Neither endpoint is stationary: the derivative is $2$ at $x=0$ and $-2e^\pi$ at $x=\pi/2$.

For $0<x<\pi/2$, use the double-angle formulae:

Dividing by $\cos^2x>0$,

Therefore $\tan x=1\pm\sqrt2$. Since $\tan x>0$,

Substitution into the curve gives the stationary point