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

1 Answer
Apr 1, 2015

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.)