Question

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. (Enter your answers as a comma-separated list.)
f(x) = sqrt(x) − (1/9)x, [0, 81]
c=?
f(0)=0
f(9)=0
So, that satisfies the theorem
f is continuous and differentiable.

So, we just have to find c such that f'(c)=0
f'(x) = 1/(2√x) - 1/9
1/(2√c) - 1/9 = 0
1/(2√c) = 1/9
2√c = 9
c = 81/4 which is in [0,81]

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