## Mean Value Theorem Questions

In this page mean value theorem questions we are going to see some practice questions of the topic Rolle's theorem.

(1) Verify Rolle's theorem for the following functions:

(i) f (x) = sin x   0 ≤ x ≤ π

(ii) f (x) = x², 0 ≤ x ≤ 1

(iii) f (x) = |x - 1|, 0 ≤ x ≤ 2

(iv) f (x) = 4 x³ - 9 x, -3/2 ≤ x ≤ 3/2

(2) Using Rolle's theorem find the points on the curve y = x² + 1,

- 2 ≤ x ≤ 2 where the tangent is parallel to x - axis.

Solution

(1) Verify Rolle's theorem for the following functions:

(i) f (x) = sin x   0 ≤ x ≤ π

If f(x) be a real valued function that satisfies the following three conditions.

 1. f(x) is defined and continuous on the closed interval [0,π] 2. f(x) is differentiable on the open interval (0,π). f (X) = sin x f (0) = sin 0        = 0f (π) = sin π         = 0f (0) = f (π)from this we come to know that the given function satisfies all the conditions of Rolle's theorem. c ∈ (0,π) we can find the value of c by using the condition f '(c) = 0.f (x) = sin xf '(x) = cos xf '(c) = cos c f '(c) = 0Cos c = 0       c = (2 n + 1) (π/2)        c = π/2,3π/2,5π/2,.............Here π/2 is the only value that is in the given interval.So the value of c = π/2   mean value theorem questions

(ii) f (x) = x², 0 ≤ x ≤ 1

If f(x) be a real valued function that satisfies the following three conditions.

 1. f(x) is defined and continuous on the closed interval [0,1] 2. f(x) is differentiable on the open interval (0,1). f (X) = x²f (0) = 0²        = 0f (1) = 1²         = 1f (0) ≠ f (1)The given function is not satisfying all the conditions of mean value theorem. So we cannot find the value of c.