`f(x) = sqrt(x) - (1/3)x, [0,9]` 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) = sqrt(x) - (1/3)x, [0,9]` 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 3 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Rolle's Theorem requires that f is continuous on closed interval (true), f is differentiable on the open interval (also true) and f(a)=f(b).

f(0) = 0 - 0 = 0, f(9) = 3 - 9/3 = 0, also true.

Then there is c from (0, 9) such that f'(c) = 0.

`f'(c) = 1/(2sqrt(c)) - 1/3,` this is zero when

`1/(2sqrt(c)) = 1/3,`  `2sqrt(c) = 3,` c = 9/4 = 2.25.

(note that f isn't differentiable at x=0 but this doesn't prevent the use of Rolle's Theorem)

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