Word Count : 520

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To verify Rolle's Theorem for the function \( f(x) = x(x - 2)e^{-x} \) on the interval \([0, 2]\), we need to check the three conditions of Rolle's Theorem: 1. Continuity of \( f(x) \) on the closed interval \([0, 2]\): The function \( f(x) = x(x - 2)e^{-x} \) is a product of continuous functions (polynomials and the exponential function), so it is continuous on \([0, 2]\). 2. Differentiability of \( f(x) \) on the open interval \((0, 2)\): Since \( f(x) \) is composed of polynomial and exponential functions, it is differentiable on \((0, 2)\). 3. Equality of the function values at the endpoints: We need to check if \( f(0) = f(2) \). Let's compute these: ### Step 1: Calculate \( f(0) \) and \( f(2) \) __________ ____ _______ _____ _______ __________ ________ ___ ___ _______ ____ ___.
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