How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [-1,1] for #f(x) = x(x^2 - x - 2)#?

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Answer 1 · 2017-02-06T13:08:33

Solve the relevant equation.

The conclusion of MVT says

"There is a #c# in #(a,b)# such that #f'(c) = (f(b)-f(a))/(b-a)#."

To find the #c#, solve the equation, discard any solutions outside of #(a,b)#.

In this case, we need to solve

#3x^2-2x-2 = (f(1)-f(-1))/(1-(-1))#.

We get #x=-1/3# or #1#.

Since #1# is not in the interval #(-1,1)#, the only value of #c# is #-1/3#.