How do you find the number c that satisfies the conclusion of the Mean Value Theorem for the function #f(x)=x^3 - 2x + 1# on the interval [0,2]?

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Answer 1 · 2016-04-19T21:06:28

Solve #f'(x) = (f(2)-f(0))/(2-0)# Discard any solution(s) outside #(0,2)#.

The conclusion of the Mean Value Theorem for function #f# on interval #[a,b]# is

there is a number #c# in #(a,b)# such that #f'(c) = (f(b)-f(a))/(b-a)#

(Alternatively such that #f(b)-f(a)=f'(c)(b-a)# which is equivalent by algebra.)

So the #c# or #c#'s that satisfy the conclusion for this function on this interval are exactly the solutions to

#f'(x) = (f(2)-f(0))/(2-0)#

that are in #(0,2)#.

Since the solutions of #3x^2-2 = 2# are #+-sqrt(4/3)#, the #c# we want is #sqrt(4/3)#, which is better written as #(2sqrt3)/3#.