Question 2 - Case Based Questions (MCQ) - Chapter 5 Class 12 Continuity and Differentiability

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Question Rolle’s Theorem: Suppose following three condition hold for function y = f(x): 1. function is defined and continuous on closed interval [a, b]; 2. exists finite derivative f ‘(x) on interval (a, b); 3. f(a) = f(b). then there exists point c(a < c < b) such that f ‘(c) = 0. Based on the above information, answer any four of the following questions. Question 1 (i) Rolle’s theorem is not applicable for the function f(x) = tan x in [0, 𝜋] because _______. (a) it is not continuous in [0, 𝜋] (b) it is differentiable in (0, 𝜋) (c) f(0) ≠ f(𝜋) (d) f(0) = f(𝜋) Since tan 𝜋/2 is not defined, tan x is not continuous at x = 𝜋/2 So, the correct answer is (A) Question 2 The value of c satisfying Rolle’s theorem for the function g(x) = sin x in [0, 𝜋] is _______. (a) 0 (b) p (c) 𝜋/2 (d) 𝜋/4 According to Rolle’s theorem, There exists a c ∈ (0, 𝜋) such that g’(x) = 0 (sin x)’ = 0 cos x = 0 ∴ x = 𝝅/𝟐 Thus, value of c = 𝝅/𝟐 So, the correct answer is (C) Question 3 The value of c satisfying Rolle’s theorem for the function h(x) = cos x in [0, 2𝜋] is _______. (a) 0 (b) 𝜋 (c) 𝜋/2 (d) 3𝜋/2 According to Rolle’s theorem, There exists a c ∈ (0, 2𝜋) such that h’(x) = 0 (cos x)’ = 0 −sin x = 0 ∴ x = 𝜋 Thus, value of c = 𝜋 So, the correct answer is (B) Question 4 The value of c satisfying Rolle’s theorem for the function p(x) = sin x + cos x in [0, 𝜋] is _______. (a) 0 (b) 𝜋 (c) 𝜋/4 (d) 𝜋/2 According to Rolle’s theorem, There exists a c ∈ (0, 𝜋) such that p’(x) = 0 (sin x + cos x)’ = 0 cos x − sin x = 0 cos x = sin x ∴ x = 𝜋/4 Thus, value of c = 𝜋/4 So, the correct answer is (C) Question 5 Rolle’s theorem is not applicable for the function f (x) = |x| in [–2, 2] because _______. (a) f (–2) ¹ f(2) (b) f(x) is not continuous in [–2, 2] (c) f(x) is not differentiable in (–2, 2) (d) None of these We know that |𝑥| is not differentiable at x = 0. So, the correct answer is (C)