To find the maximum value of the function (f(x) = \sin x - \cos x) over the interval ([0, \pi]), follow these steps:

Step 1: Compute the derivative

The derivative of (f(x)) is:
[f'(x) = \cos x + \sin x]

Step 2: Find critical points

Set (f'(x) = 0):
[ \cos x + \sin x = 0 \implies \tan x = -1]

In the interval ([0, \pi]), (\tan x = -1) at (x = \frac{3\pi}{4}) (second quadrant, where (\sin) is positive and (\cos) is negative).

Step 3: Evaluate (f(x)) at critical points and endpoints

• At (x=0): (f(0) = \sin 0 - \cos 0 = -1)
• At (x=\pi): (f(\pi) = \sin \pi - \cos \pi = 1)
• At (x=\frac{3\pi}{4}): (f\left(\frac{3\pi}{4}\right) = \sin\left(\frac{3\pi}{4}\right) - \cos\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} - \left(-\frac{\sqrt{2}}{2}\right) = \sqrt{2})

Conclusion

The maximum value is (\sqrt{2}).

Answer: (\boxed{\sqrt{2}})

作者声明:本文包含人工智能生成内容。