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

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

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Textbook Question

Chapter 4, 4.2 - Problem 10 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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mathace | (Level 3) Assistant Educator

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Given: `f(x)=x^3-3x+2,[-2,2]`

The function is continuous on the closed interval [-2,2].

The function is differentiable on the open interval (-2,2).

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

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

`[f(2)-f(-2)]/(2-(-2))=(4-0)/(2-(-2))=1`

`3c^2-3=1`

`3c^2=4`

`c^2=4/3`

`c=[+-2sqrt(3)]/3=+-1.155`

The answers are `+-1.155`  . Both c values are in the interval [-2,2].

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