Question

How do you find the number c that satisfies the conclusion of the Mean Value Theorem for the function `f(x)=x^3 - 2x + 1` on the interval [0,2]?

Answer 1

Solve `f'(x) = (f(2)-f(0))/(2-0)` Discard any solution(s) outside `(0,2)`.

Explanation:

The conclusion of the Mean Value Theorem for function `f` on interval `[a,b]` is

there is a number `c` in `(a,b)` such that `f'(c) = (f(b)-f(a))/(b-a)`

(Alternatively such that `f(b)-f(a)=f'(c)(b-a)` which is equivalent by algebra.)

So the `c` or `c`'s that satisfy the conclusion for this function on this interval are exactly the solutions to

`f'(x) = (f(2)-f(0))/(2-0)`

that are in `(0,2)`.

Since the solutions of `3x^2-2 = 2` are `+-sqrt(4/3)`, the `c` we want is `sqrt(4/3)`, which is better written as `(2sqrt3)/3`.