The c has only one number is c = 81/4 that satisfy the conclusion of rolle's theorem.
The three hypotheses of Rolle's theorem are:
(a) f(x) is continuous on [a,b]
(b) f(x) is differentiable on (a,b)
(c) f(a) = f(b)
i) If you graph f(x), you can see that it is continuous that is no breaks on the interval [0,81]
ii) f(x) is differentiable on (0,81)
f '(x) = 1/(2√x) - 1/9 which exists on the open interval (0,81)
iii) f(0) = f(81) = √81 - 81/9 = 9-9 = 0
Rolle's Conclusion:
f'(c) = 0
f'(c) = 1/(2√c) -1/9 = 0
(9 - 2√c)/(18√c) = 0
√c = 9/2
c = 81/4
There is one number c = 81/4, which satisfies f '(c) = 0.
Therefore the c has only one number is c = 81/4 that satisfy the conclusion of rolle's theorem.
Learn more about the rolle's theorem here:
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Full 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) =x − 1/9 x, [0, 81].
