Question

Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b].

f(x)=√(1-x), [-24,1]

Answer 1

The Mean Value Theorem cannot be applied to f(x) = √(1-x) on the interval [-24, 1] because the function is not continuous over the entire interval, as the square root function is undefined for x < 0.

Step-by-step explanation:

To determine whether the Mean Value Theorem can be applied to the function f(x) = √(1-x) on the closed interval [a, b] = [-24, 1], we must check if the function satisfies the theorem's two main criteria:

1. The function must be continuous on the closed interval [a, b].
2. The function must be differentiable on the open interval (a, b).

If we consider f(x) = √(1-x), the function is continuous wherever it is defined since it is a composition of continuous functions (the square root function and the linear function). However, the square root function is not defined for negative numbers, which means that f(x) is not defined over the entire interval from -24 to 1, specifically it is undefined for any x < 0. Hence, it is not continuous on the closed interval [-24, 1], and the first criterion is not met.

The Mean Value Theorem cannot be applied to f(x) = √(1-x) on [-24, 1] because the function is not continuous over the entire interval. Even if the function were continuous, we would also need to check its differentiability, but since the continuity criterion has already failed, the theorem does not apply.