How do you determine whether #f(x) = sqrt( 2x + 3 ) -1# satisfies the hypotheses of the Mean Value Theorem on the interval [3,11] and find all value(s) of c that satisfy the conclusion of the theorem?

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Answer 1 · 2015-05-07T15:00:45

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(2x+3)-1# , #a=3# and #b=11#.

This function is continuous on its domain, #[-3/2, oo)#, so it is continuous on #[3, 11]#

#f'(x)=1/(sqrt(2x+3))# which exists (and is real) for all #x > -3/2# (for all #x in (-3/2,oo)#,

So #f# is differentiable on #(3, 11)#

Therefore, #f(x) = sqrt(2x+3)-1# satisfies the hypotheses of the Mean Value Theorem on the interval #[3, 11]#.

To do the additional algebra problem,

find #(f(11)-f(3))/(11-3)#

Set #f'(x) = (f(11)-f(3))/(11-3)# and solve.

Any solution in #(3, 11)# is one of the #c#'s that satisfy the conclusion.

(If you need help with the arithmetic or the algebra, let us know.)