Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

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]

Answer 1

The three hypotheses of Rolle's theorem are:

(i) f(x) is continuous on [a,b]

(ii) f(x) is differentiable on (a,b)

(iii) f(a) = f(b)

1) If you graph f(x), you can see that it is continuous, i.e. no breaks on the interval [0,81]

2) f(x) is differentiable on (0,81)

f '(x) = 1/(2√x) - 1/9 which exists on the open interval (0,81)

3) 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 only one number c = 81/4 which satisfies f '(c) = 0