Using the principle of the mean-value theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem for #f(x) = 5(1 + 2x)^(1/2)# in the interval [1,4]?

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Answer 1 · 2015-10-19T01:13:42

Solve the equation that is the conclusion to the Mean Value Theorem. Choose only the solutions in the interval #(1,4)# as the conclusion says.

Find #f'(x)#, then solve

#f'(x) = (f(4)-f(1))/(4-1)#.

The algebra is tedious.

#f(4) = 15#, #f(1) = 5sqrt3#

#f'(x) = 5/sqrt(1+2x)#

So we need to solve:

#5/sqrt(1+2x) = (15-5sqrt3)/3#. Which is equivalent to

#1/sqrt(1+2x) = (3-sqrt3)/3#

So, we need #sqrt(1+2x) = 3/(3-sqrt3) = (3+sqrt3)/2#

#1+2x = (3+sqrt3)^2/4 = (6+3sqrt3)/2#

So #x = (4+3sqrt3)/4 = 1 + (3sqrt3)/4#.

Note that, since #3sqrt3 = sqrt 27 < 12#, this solution is in the interval #(1,4)#