Question

Which of the following functions satisfies the conditions of mean value theorem over the indicated interval?

1) f(x) = x² - 4x + 3, [1, 3]
2) f(x) = 2x + 1, [-1, 1]
3) f(x) = 3x² + 2x - 1, [-2, 2]
4) f(x) = 4x³ - 2x² + 1, [0, 1]

Answer 1

All four provided functions are polynomials, which means they are continuous everywhere and differentiable everywhere. Consequently, each function satisfies the Mean Value Theorem over their respective intervals.

Step-by-step explanation:

The Mean Value Theorem (MVT) states that for any function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one c in the interval (a, b) such that f'(c) is equal to the function's average rate of change over [a, b]. Mathematically, this is expressed as:

f'(c) = (f(b) - f(a)) / (b - a).

To determine which functions satisfy the conditions of the Mean Value Theorem, we must check for continuity on the closed interval and differentiability on the open interval. Let's analyze each function:

1. f(x) = x² - 4x + 3 on [1, 3]
2. f(x) = 2x + 1 on [-1, 1]
3. f(x) = 3x² + 2x - 1 on [-2, 2]
4. f(x) = 4x³ - 2x² + 1 on [0, 1]

All provided functions are polynomials, which are continuous and differentiable on the entire real line. Therefore, all four functions satisfy the continuity and differentiability conditions of the Mean Value Theorem over their indicated intervals.