题目

Problem

Use proof by contradiction to prove that the curve with equation

y=2x+x^3+\cos x

has no stationary points.

(4)

题目中文翻译

用反证法证明方程

y=2x+x^3+\cos x

所表示的曲线没有驻点。

解答

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按反证法，先假设曲线存在驻点，因此在某个实数 $x$ 处导数为 $0$ 。求导并整理后，会得到 $\sin x=3x^2+2$ ；但右边至少为 $2$ ，与正弦函数的值域矛盾。

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Assume, for a contradiction, that the curve has a stationary point for some real value of $x$ . Then

\frac{\mathrm{d}y}{\mathrm{d}x}=0.

Differentiating,

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

Therefore, at the assumed stationary point,

\begin{align*} 2+3x^2-\sin x=&\,0\\ \sin x=&\,3x^2+2. \end{align*}

However, $x^2\geq0$ , so

3x^2+2\geq2,

whereas $\sin x\leq1$ for every real $x$ . This is a contradiction.

Hence the original assumption is false, and the curve has

\boxed{\text{no stationary points}}.