Question

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

f(x)=x / 1,[1,2]
A Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem.
B Yes, f is continuous on [1,2] and differentiable on (1,2).
C No, f is not continuous on [1,2].
D No, f is continuous on [1,2] but not differentiable on (1,2).
E There is not enough information to verify if this function satisfies the Mean value Theorem.
If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem.
C = _______

Answer 1

Yes, the function satisfies the hypotheses of the Mean Value Theorem on the given interval [1,2]. The number c that satisfies the conclusion of the Mean Value Theorem is any number in the interval (1,2).

Step-by-step explanation:

The Mean Value Theorem states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists at least one number c in (a,b) such that f'(c) = (f(b) - f(a))/(b - a).

In the given function f(x) = x / 1, the function is continuous on the interval [1,2] and differentiable on the open interval (1,2). Therefore, the function satisfies the hypotheses of the Mean Value Theorem.

In order to find the number c that satisfies the conclusion of the Mean Value Theorem, we need to find the derivative of f(x) and solve for c in the equation f'(c) = (f(b) - f(a))/(b - a). Given that f(x) = x / 1 and the interval [1,2], we have f'(x) = 1 / 1. Substituting the values into the equation, we get 1/1 = (2/1 - 1/1)/(2 - 1). Simplifying, we get 1 = 1/1, which is true for any number c in the interval (1,2).

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