JEE Advanced2014MathematicsContinuity and DifferentiabilityActual
Let f : a , b → 1 , ∞ be a continuous function and let g : R → R be defined as g x = 0 i f x < a , ∫ a x f t d t i f a ≤ x ≤ b , ∫ a b f t d t i f x > b . Then
Options
- Ag x is continuous but not differentiable at a
- Bg x is differentiable on R
- Cg x is continuous but not differentiable at b
- Dg x is continuous and differentiable at either a or b but not both
Correct answer
A. g x is continuous but not differentiable at a
Step-by-step solution
Since f x ≥ 1 ∀ x ∈ a , b for g x LHD at x = a is zero And RHD at x = a = lim x → a + ∫ a x f t d t - 0 x - a = lim x → a + f x ≥ 1 Hence g x is not differentiable at x = a Similarly LHD at x = b is greater than 1 g x is not differentiable at x = b