How do you determine all values of c that satisfy the mean value theorem on the interval [0, 2] for #y = x^3 + x - 1#?

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Answer 1 · 2017-12-19T22:53:22

See below.

Let #f(x) = x^3+x-1#

The conclusion of the mean value theorem says:

there is a #c# in #(a,b)# with #f'(c) = (f(b)-f(a))/(b-a)#.

To find that #c# (or those #c#'s, find the equation and solve it.

So if you want to actually find the #c# mentioned in the conclusion to the theorem, then you need to solve the equation.

In this case solve

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

Discard any solutions outside #(0,2)#

You should get #c = (2sqrt3)/3#