Find the maximum and minimum values of the function $f(x) = 5 - \sin x$.

1. Problem Description

Find the maximum and minimum values of the function $f(x) = 5 - \sin x$.

2. Solution Steps

We know that the range of the sine function is $-1 \le \sin x \le 1$. Then we can use this to find the range of $f(x) = 5 - \sin x$.

We have:

$-1 \le \sin x \le 1$

Multiplying by $-1$ reverses the inequality signs:

$1 \ge -\sin x \ge -1$

Adding $5$ to all sides gives:

$1 + 5 \ge 5 - \sin x \ge -1 + 5$

$6 \ge 5 - \sin x \ge 4$

Thus $4 \le f(x) \le 6$.

The maximum value of $f(x)$ is $6$, which occurs when $\sin x = -1$, for example $x = \frac{3\pi}{2}$.

The minimum value of $f(x)$ is $4$, which occurs when $\sin x = 1$, for example $x = \frac{\pi}{2}$.

3. Final Answer

The maximum value of $f(x)$ is $6$, and the minimum value is $4$.