Final answer:
In the given function f(x) = √2 - x² on the interval [-√2, 0], the number that satisfies the conclusion of the Mean Value Theorem is c = -1/2. This is found by taking the derivative of the function, setting it equal to the average rate of change over the interval, and solving for c.
Step-by-step explanation:
You've asked to find all numbers c, with √2 < c < 0, that satisfy the conclusion of the Mean Value Theorem for the function f(x) = √2 − x2 on the interval [√2, 0]. The Mean Value Theorem states that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one number c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
First, we check that the function is continuous and differentiable on the given interval. Since the function is a polynomial under a square root and the interval is within the domain of f(x), f(x) is continuous and differentiable on the interval [√2, 0] except at the endpoints.
Next, we find the derivative of f(x), which is f'(x) = -2x. Now, we evaluate f(x) at the endpoints:
- f(-√2) = √2 - (√2)2 = √2 - 2 = 0
- f(0) = √2 - (0)2 = √2
Applying the Mean Value Theorem, we set f'(c) equal to the average rate of change from f(-√2) to f(0):
-2c = (√2 - 0) / (0 - (-√2))
This simplifies to -2c = √2 / √2 = 1. Therefore, c = -1/2.
Thus, c = -1/2 is the number that satisfies the conclusion of the Mean Value Theorem for the function f(x) on the interval [√2, 0].