Find all numbers c that satisfy the conclusion of Rolle's Theorem.  f(x) = x^3 − x^2 − 2x + 8     interval= [0, 2]

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`f(0)=8`

`f(2)=8-4-4+8=8`

f is continuous, and differentiable on [0,2].

therefore by Rolle's Theorem, `EEc in(0,2) | f'(c)=0`

 

Let's find c.

`f'(x)=3x^2-2x-2`

using the quadratic formula `f'(x)=0` iff `x=(2+-sqrt(4+4*3*2))/(6)`

`x=(1+-sqrt(7))/3`

Since `c in(0,2)` , `c=(1+sqrt(7))/3`

 

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