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)+3f(x)=2x+3?

1 Answer
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.