Verity that the Mean Value Theorem can be applied to the function f (t) = 2% on the interval (0, 1). Then find the value of e in the interval that satisfies the conclusion of the Mean Value Theorem. Enter the exact answer a ab va a M sin(a) 9 TIT

Sol58:

To verify that the Mean Value Theorem can be applied to the function f(t) = 2t on the interval (0,1), we need to check if the function is continuous on the interval (0,1) and differentiable on the interval (0,1).

The function f(t) = 2t is continuous on the interval (0,1) as it is a polynomial function.

The function f(t) = 2t is also differentiable on the interval (0,1) as its derivative f\\\'(t) = 2 is a constant function.

Therefore, the Mean Value Theorem can be applied to the function f(t) = 2t on the interval (0,1).

By the Mean Value Theorem, there exists a value e in the interval (0,1) such that:

f\\\'(e) = (f(1) - f(0))/(1-0)

2 = 2/(1-0)

2 = 2

Thus, any value e in the interval (0,1) satisfies the conclusion of the Mean Value Theorem.