Question

How do you determine whether the Mean Value Theorem can be applied to `f(x) = sqrt (x-7)` on the interval [11,23]?

Answer 1

The Mean Value Theorem has two hypotheses:

H1 : `f` is continuous on the closed interval `[a,b]`

H2 : `f` is differentiable on the open interval `(a,b)`.

In this question, `f(x) = sqrt(x-7)` , `a=11` and `b=23`.

We can apply the Mean Value Theorem is both hypotheses are true.

This function is continuous on its domain, `[0, oo)`, so it is continuous on `[11, 23]`

`f'(x)=1/(2sqrt(x-7))` which exists for all `x >7` (for all `x in (7,oo)`,

So `f` is differentiable on `(11, 23)`