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

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Answer 1 · 2015-04-28T14:12:43

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)#