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1. Verify that the function satisfies the three hypotheses of Rolle’s Thoerem on the given interval. Then find all the numbers c that satisfy the conclusion of Rolle’s Theorem. (Enter your answers as a comma-seperated list.) F(x) = 2 – 24x + 3x^2, [3,5]

2. Verify that the function satisfies the three hypotheses of Rolle’s Thoerem on the given interval. Then find all the numbers c that satisfy the conclusion of Rolle’s Theorem. (Enter your answers as a comma-seperated list.) F(x) = x^3 – x^2 – 20x + 4, [0,5]

3. Consider the following function. f(x) = 9 – x^2/3

Find f(-27) and f(27)

Based off of this information, what conclusions can be made about Rolle’s Theorem?

4. 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-seperated list.) f(x) = cos 3x, [pi/12, 7pi/12]

5. Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

F(x) = 4x^2 – 2x + 3, [0, 2]

If it satisfies the hypotheses, find all the numbers c that satisfy the conclusion of the mean value theorem. (Enter your answers as a comma-seperated list. If it does not satisfy the hypotheses, enter DNE)

6. Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

F(x) = ln x, [1, 3]

If it satisfies the hypotheses, find all the numbers c that satisfy the conclusion of the mean value theorem. (Enter your answers as a comma-seperated list. If it does not satisfy the hypotheses, enter DNE)

7. If f(2) = 9 and f’(x) >= 2 for 2<= x <= 5, how small can f(5) possibly be?

8. Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

F(x) = 2x^2 + 5x +3, [-1, 1]

If it satisfies the hypotheses, find all the numbers c that satisfy the conclusion of the mean value theorem. (Enter your answers as a comma-seperated list. If it does not satisfy the hypotheses, enter DNE)