SECTION 4.2 THE MEAN VALUE THEOREM - I 4.2 THE MEAN VALUE THEOREM A Click here for answers. S Click here for solutions. 1-4 - Verify that the function satisfies the three hypotheses of 9. f(x) = 1/x; [1, 2] Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. 10. f(x) = Vx; [1, 4] 1. f (x ) = x3 - x, [-1, 1] I1. f(x) = 1+ Vx - 1; [2, 9] 2. f ( x ) = x3 + x2 - 2x+ 1, [-2, 0] 2 3. f(x ) = cos 2x, [0, TT] 12. Verify that the function f(x) = x - 6x3 + 4x - 1 satisfies the 1 4. f(x) = sin x + cos x, [0, 2or] hypotheses of the Mean Value Theorem on the interval [0, 1]. Then use a graphing calculator or CAS to find, correct to two decimal places, the numbers c that satisfy the conclusion of the 5-11 - Verify that the function satisfies the hypotheses of the Mean Value Theorem. Mean Value Theorem on the given interval. Then find all numbers c 13. Show that the equation x' + 10x + 3 = 0 has exactly one real that satisfy the conclusion of the Mean Value Theorem. root. 15. f(x) = x2 - 4x + 5, [1, 5] 14. Show that the equation 3x - 2 + cos( Tx/2) = 0 has exactly one real root. 1 6. f ( x) = x3 - 2x + 1, [-2, 3 ] 15. Show that the equation x' - 6x + c = 0 has at most one root 7. f(x) = 1 - x2, [0, 3] in the interval [- 1, 1]. 8. f(x) = 2x3 + x2 - x - 1, [0, 2] 16. Suppose f is continuous on [2, 5] and 1