Question

. Find the value of c in the Mean Value Theorem: f(x)=x2+3x+2 on the interval [a,b], you need to find the derivative f′(c) that equals the average rate of change of f over that interval.

Answer 1

To find the value of c in the Mean Value Theorem for f(x) = x^2 + 3x + 2, we need to find f'(c) that equals the average rate of change of f over the interval [a, b].

Step-by-step explanation:

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that the derivative of the function at c, denoted as f′(c), is equal to the average rate of change of the function over the interval [a, b].

In this case, the function is f(x) = x^2 + 3x + 2. To find the value of c, we need to find the derivative f′(c) and set it equal to the average rate of change of f over the interval [a, b].

Step 1: Find the derivative of f(x) using the power rule of differentiation: f′(x) = 2x + 3.

Step 2: Set f′(c) equal to the average rate of change of f over [a, b], which is (f(b) - f(a))/(b - a).

So, 2c + 3 = (f(b) - f(a))/(b - a).

Step 3: Substitute the values of a, b, f(a), and f(b) into the equation and solve for c.