How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [3, 5] for #f(x)=2sqrt(x)+3#?

1 Answer
Jim H
Nov 22, 2016

Solve #f'(x) = (f(5)-f(3))/(5-3)# on the interval #(3,5)#.

Explanation:

#f'(x) = 1/sqrtx# and

#(f(5) -f(3))/(5-3) = sqrt5-sqrt3#.

Solve:

#1/sqrtx = sqrt5-sqrt3#.

We get

#x= (1/(sqrt5-sqrt3))^2#

# = 1/(8-2sqrt15)#

# = (4+sqrt15)/2#

Since #sqrt16# is a bit less than #4#, this # (4+sqrt15)/2# is a bit less than #(4+4)/2 = 4# and is in the interval #(3,5)#.

As an alternative at the end we could point out that Since #f# satisfies the hypotheses of MVT on #[3,5}#, it must satisfy the conclusion. Furthermore with only one candidate for #c#, this must be the #c# mentioned in the conclusion of MVT.

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