(a) 证明方程 $4x - 2\sin x - 1 = 0$（其中 $x$ 以弧度为单位）在区间 $[0.2, 0.6]$ 内有一个根 $\alpha$。

(b) 从区间 $[0.2, 0.6]$ 开始，使用区间二分法两次，求宽度为 $0.1$ 且包含 $\alpha$ 的区间。

令 $f(x)=4x-2\sin x-1$。计算区间两端的函数值；由于 $f$ 连续且两端函数值异号，根据介值定理，区间内至少有一个根。

Let

Then, using radians,

The function $f$ is continuous, and $f(0.2)$ and $f(0.6)$ have opposite signs. Therefore, by the intermediate value theorem, there is a root $\alpha$ in

从 $[0.2,0.6]$ 开始，第一次取中点 $0.4$，根据函数值符号把区间缩为 $[0.4,0.6]$；第二次取中点 $0.5$，再次利用异号端点，把区间缩为宽度 $0.1$ 的区间。

For the first bisection, the midpoint of $[0.2,0.6]$ is $0.4$, and

Since $f(0.4)<0$ and $f(0.6)>0$, the root lies in $[0.4,0.6]$.

For the second bisection, the midpoint of $[0.4,0.6]$ is $0.5$, and

Since $f(0.4)<0$ and $f(0.5)>0$,

This interval has width $0.1$.