`f(x) = x^2 - 8x + 5, [2,6]` Determine whether Rolle’s Theorem can be applied to `f` on the closed interval `[a, b]`. If Rolle’s Theorem can be applied, find all values of `c` in the open...

`f(x) = x^2 - 8x + 5, [2,6]` Determine whether Rolle’s Theorem can be applied to `f` on the closed interval `[a, b]`. If Rolle’s Theorem can be applied, find all values of `c` in the open interval.

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Chapter 3, 3.2 - Problem 10 - Calculus of a Single Variable (10th Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Given: `f(x)=x^2-8x+5,[2,6].`

Rolle's Theorem can be applied because the function f(x) is a continuous polynomial on the closed interval [2,6] and differentiable on the open interval (2,6)., and f(2)=f(6)=-7.

`f'(c)=[f(b)-f(a)]/(b-a)=[-7-(-7)]/[6-2]=0/4=0`

`f'(x)=2x-8`

`f'(c)=2c-8=0`

`2c=8`

`c=4`

The value c=4 is within the interval [2,6].

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