`f(x) = x^3 - x^2 - 6x + 2, [0,3]` Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers `c` that satisfy the conclusion of...

`f(x) = x^3 - x^2 - 6x + 2, [0,3]` Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers `c` that satisfy the conclusion of Rolle’s Theorem.

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Chapter 4, 4.2 - Problem 2 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Hypotheses 1, 2: f is contionuous on [a, b] and is differentiable on (a, b). It is obvious for a polynomial function.

Hypothesis 3: f(a)=f(b), here f(a)=f(0)=2, f(b)=f(3)=27-9-18+2=2.

So the theorem may be applied, there is at least one c in (0, 3) such that f'(c)=0.

Here` f'(x) = 3x^2 - 2x - 6,` f'(x)=0 for

`x_(1,2) = (1 +- sqrt(1+18))/3 = (1 +- sqrt(19))/3.`

Root with "-" is <0 and not in (0, 3), root with "+" is in (0, 3) and approx 1.79.

The answer: `c=(1+sqrt(19))/3 approx 1.79` .

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