Question

Given the function `f(x) = 2x^2 − 3x + 1`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,2] and find the c?

Answer 1

To satisfy the hypotheses of the Mean Value Theorem a function must be continuous in the closed interval and differentiable in the open interval.

Explanation:

As `f(x) = 2x^3-3x+1` is a polynomial, it is continuous and has continuous derivatives of all orders for all real `x`, so it certainly satisfies the hypotheses of the theorem.

To find the value of `c`, calculate the derivative of `f(x)` and state the equality of the Mean Value Theorem:

`(df)/(dx)= 4x-3`

`(f(b)-f(a))/(b-a) = f'(c)`

`f(x)_(x=0) = 1`

`f(x)_(x=2) = 3`

Hence:

`(3-1)/2 = 4c-3`

and `c=1`.