Verify Rolles theorem for the function: `f(x)=e^x(sinx-cosx)` on `[pi//4,\ 5pi//4]` .

Explanation

To verify Rolle's Theorem for the function \( f(x) = e^x(\sin x - \cos x) \) on the interval \([ \frac{\pi}{4}, \frac{5\pi}{4} ]\), we will follow these steps:

### Step 1: Check the continuity and differentiability
Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point \( c \) in the open interval where the derivative \( f'(c) = 0 \).

1. **Continuity**: The function \( f(x) = e^x(\sin x - \cos x) \) is a product of continuous functions (exponential, sine, and cosine functions). Therefore, \( f(x) \) is continuous on \([ \frac{\pi}{4}, \frac{5\pi}{4} ]\).
2. **Differentiability**: Since \( f(x) \) is continuous and consists of elementary functions, it is also differentiable on the open interval \( ( \frac{\pi}{4}, \frac{5\pi}{4} ) \).

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