图中显示了 $y=\mathrm{e}^{\sin2x}\cos4x$ 在 $0\le x\le\frac14\pi$ 上的图像及其极大值点 $M$。

求 $M$ 的 $x$ 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。

用乘积与链式法则求导。因指数因子恒正，驻点方程可约去它，再用二倍角化为关于 $\sin2x$ 的二次方程。

Differentiating,

At the maximum, the derivative is zero. Since $e^{\sin2x}>0$,

Using $\sin4x=2\sin2x\cos2x$,

For an interior point $0<x<\pi/4$, $\cos2x>0$. Hence, using $\cos4x=1-2\sin^22x$,

Writing $s=\sin2x$,

Only $s=-1+\sqrt6/2$ belongs to $(0,1)$. Therefore

As $x$ increases through this value, $1-2s^2-4s$ changes from positive to negative, so the stationary point is a maximum.