Question

Using the principle of the mean-value theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem `f(x)=x^3-2x^2`; [0, 2]?

Answer 1

The conclusion of the Mean Value Theorem for `f` on `[a,b]` says:

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

(The theorem make no guarantees about our ability to find the value(s) of `c`)

Find the value(s) of `c` does not use the Mean Value Theorem, it uses the derivative and some algebra.

For `f(x) = x^3-2x^2` on `[0,2]`, the conclusion of MVT says:

there is a `c` in `(0,2)` such that `f'(c) = (f(2)-f(0))/(2-0)`

To find the `c` (or `c`'s) find `f'(x)`, do the arithmetic on the right and solve the resulting equation. In this case, solve

`3x^2-4x = 0`

The solutions are `0` and `4/3`.

The `c` mentioned in MVT must be in `(0,2)`, so `4/3` is the only value of `c`