1. Verify Rolles theorem for the function f (x) = x2 + 2x 8, x [ 4, 2]. 2. Examine if Rolles theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolles theorem from these example? (i) f (x) = [x] for x [5, 9] (ii) f (x) = [x] for x [ 2, 2] (iii) f (x) = x2 1 for x [1, 2] 3. If f : [ 5, 5] R is a differentiable function and if f (x) does not vanish anywhere, then prove that f ( 5) f (5). 4. Verify Mean Value Theorem, if f (x) = x2 4x 3 in the interval [a, b], where a = 1 and b = 4. 5. Verify Mean Value Theorem, if f (x) = x3 5x2 3x in the interval [a, b], where a = 1 and b = 3. Find all c (1, 3) for which f (c) = 0. 6. Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
Question 1:Verify Rolle’s theorem for the function f (x) = x2 + 2x – 8, x Î [– 4, 2].
Question 2:Examine if Rolle’s theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s theorem from these example? (i) f (x) = [x] for x Î [5, 9] (ii) f (x) = [x] for x Î [– 2, 2] (iii) f (x) = x2 – 1 for x Î [1, 2]
Question 3:If f : [– 5, 5] → R is a differentiable function and if f ¢(x) does not vanish anywhere, then prove that f (– 5) ¹ f (5).
Question 4:Verify Mean Value Theorem, if f (x) = x2 – 4x – 3 in the interval [a, b], where a = 1 and b = 4.
Question 5:Verify Mean Value Theorem, if f (x) = x3 – 5x2 – 3x in the interval [a, b], where a = 1 and b = 3. Find all c Î (1, 3) for which f ¢(c) = 0.
Question 6:Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
