Question 3
The value of c in Rolle’s theorem for the function f (x) = x3 – 3x in the interval [0,√3] is
(A) 1 (B) −1
(C) 3/2 (D) 1/3
𝑓 (𝑥)= 𝑥3 −3𝑥 , 𝑥 ∈ [0,√3]
First, we will check if the
conditions of Rolle’s theorem
are satisfied
Conditions of Rolle’s theorem
𝑓(𝑥) is continuous at {𝑎 , 𝑏}
𝑓(𝑥) is derivable at (𝑎 , 𝑏)
𝑓(𝑎) = 𝑓(𝑏)
If all 3 conditions satisfied then there exist some c in (𝑎 , 𝑏)
such that 𝑓′(𝑐) = 0
Condition 1
We need to check if 𝑓(𝑥)=𝑥3 −3𝑥 is continuous at [𝟎,√𝟑]
Since 𝑓(𝑥)=𝑥3 −3𝑥 is a polynomial
& Every polynomial function is continuous for all 𝑥 ∈𝑅
∴ 𝑓(𝑥)=𝑥3 −3𝑥 is continuous at 𝑥∈[0,√3]
Condition 2
We need to check if 𝑓(𝑥)=𝑥3 −3𝑥 is differentiable at (0,√3)
𝑓(𝑥) =𝑥3 −3𝑥 is a polynomial
& Every polynomial function is differentiable for all 𝑥 ∈𝑅
∴ 𝒇(𝒙) is differentiable at (0,√3)
Condition 3
Finding 𝒇(𝟎)
𝑓(𝑥) = 𝑥3 −3𝑥
𝑓(𝟎) = 03 −3(0)
= 0
Finding 𝒇(𝝅)
𝑓(𝑥) = 𝑥3 −3𝑥
𝑓(√𝟑) = (√3)3 −3(√3)
= 3(√3)−3(√3)
= 0
Hence, 𝒇(𝟎) = 𝒇(√𝟑)
Now,
𝑓(𝑥) = 𝑥3 −3𝑥
𝑓^′ (𝑥) = 3𝑥^2−3
𝒇^′ (𝒄) = 𝟑𝒄^𝟐−𝟑
Since all three condition satisfied
𝒇^′ (𝒄) = 𝟎
3𝑐^2−3= 0
3𝑐^2 = 3
𝑐^2 = 3/3
𝑐^2 = 1
c = 1
Since value of c = 1 ∈(𝟎,√𝟑)
So, the correct answer is (A)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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