This is simply the intermediate value theorem, plus knowing that a function's derivative is continuous wherever it is defined.
The intermediate value theorem says that g(x) is continuous on [a,b] then for any value d in [f(a), f(b)] there exist a value c in [a,b] such that f(c) = d.
In plain terms, the function g(x) attains each value between g(a) and g(b) somewhere in the interval [a,b].
Answers & Comments
This is simply the intermediate value theorem, plus knowing that a function's derivative is continuous wherever it is defined.
The intermediate value theorem says that g(x) is continuous on [a,b] then for any value d in [f(a), f(b)] there exist a value c in [a,b] such that f(c) = d.
In plain terms, the function g(x) attains each value between g(a) and g(b) somewhere in the interval [a,b].
Take g(x) to be f'(x) and you're done.