Let f(x)=x^(3)+x-1. Find ech number c in (1,2) that satisfies the conclusion of the Mean Value Theorem.
To apply the Mean Value Theorem to the function f(x) = x^3 + x - 1 over the interval (1, 2), we need to check if the function satisfies the two conditions of the theorem:
1. The function f(x) must be continuous on the closed interval [1, 2].
2. The function f(x) must be differentiable on the open interval (1, 2).
Let's first check if f(x) is continuous on [1, 2]. The function is a polynomial, and all polynomials are continuous for all values of x. So, f(x) is continuous on [1, 2].
Next, let's check if f(x) is differentiable on (1, 2). Since f(x) is a polynomial, it is differentiable for all values of x. Therefore, f(x) is differentiable on (1, 2).
Since f(x) satisfies both conditions, we can conclude that there exists at least one number c in (1, 2) that satisfies the conclusion of the Mean Value Theorem.
To find this value of c, we need to find the derivative of f(x) and solve for c.
The derivative of f(x) is: f'(x) = 3x^2 + 1.
Now, we can solve the equation f'(c) = (f(b) - f(a))/(b - a), where a = 1 and b = 2.
Substituting the values, we get: 3c^2 + 1 = ((2^3 + 2 - 1) - (1^3 + 1 - 1))/(2 - 1)
Simplifying, we have: 3c^2 + 1 = (7 - 1)/1
Further simplifying, we get: 3c^2 + 1 = 6
Subtracting 1 from both sides, we have: 3c^2 = 5
Dividing both sides by 3, we obtain: c^2 = 5/3
Taking the square root of both sides, we get: c = ±√(5/3)
Since we are looking for a number c in the interval (1, 2), we discard the negative square root.
Therefore, the number c in (1, 2) that satisfies the conclusion of the Mean Value Theorem is c = √(5/3).