99 Percentile Qs Bank for JEE MainMathematicsContinuity and Differentiability
Let, f x = sin x and g x = max f ( t ) , 0 ≤ t ≤ x   ,   0 ≤ x ≤ π 1 - cos x 2 , x > π   . Then g x is
Options
- Adifferentiable for all x ∈ R
- Bdifferentiable for all x ∈ R - π
- Cdifferentiable for all x ∈ 0 , ∞
- Ddifferentiable for all x ∈ 0 , ∞ - π
Correct answer
C. differentiable for all x ∈ 0 , ∞
Step-by-step solution
Given, f x = sin x and g x = max f ( t ) , 0 ≤ t ≤ x ,   0 ≤ x ≤ π 1 - cos x 2 , x > π   We know, cos 2 A = 1 - 2 sin 2 A ∴ g x = max f ( t ) , 0 ≤ t ≤ x ,   0 ≤ x ≤ π sin 2 x 2 ,   x > π   For, 0 ≤ x ≤ π 2 , sin x is an increasing function, therefore , g x = m a x f t = sin x ,   0 ≤ x ≤ π 2 . ⇒ g x = sin x , f π 2 = 1 , sin 2 x 2 , 0 ≤ x ≤ π