Question

How do you determine whether the mean value theorem applies to `f(x)=3x-x^2`?

Answer 1

The hypotheses of the Mean Value Theorem are:

(1) `f` is continuous on closed interval `[a,b]` and

(2) `f` is differentiable on open interval `(a, b)`.

`f(x) = 3x-x^2` is a polynomial. It is continuous on every closed interval.

`f'(x) = 3-2x` exists for all `x` on any interval `(a, b)`. (In fact, we could also say that `f` is differentiable because it is a polynomial.)

Therefore for any interval `[a, b]`, the hypotheses of the mean value theorem are satisifed.

(The pedantic logician in me wants to mention that the MVT is an " if ... then ___" statement. As such it applies (in some sense) to every function. But that's not what your teacher and textbook mean. They want to know whether the hypotheses are true.)