JEE Main202127 Jul 2021Evening ShiftMathematicsContinuity and DifferentiabilityActual
Let f : 0 , ∞ → 0 , 3 be a function defined by f x = max sin t : 0 ≤ t ≤ π , x ∈ 0 , π 2 + cos x , x > π . Then which of the following is true ?
Options
- Af is continuous everywhere but not differentiable exactly at one point in 0 , ∞
- Bf is differentiable everywhere in 0 , ∞
- Cf is not continuous exactly at two points in 0 , ∞
- Df is continuous everywhere but not differentiable exactly at two points in 0 , ∞
Correct answer
B. f is differentiable everywhere in 0 , ∞
Step-by-step solution
Graph of max sin t : 0 ≤ t ≤ x in x ∈ 0 , π & the graph of y = 2 + cos x for x ∈ π , ∞ So graph of f x = sin x , 0 ≤ x ≤ π 2 1 , π 2 ≤ x ≤ π 2 + cos x , x > π f x is differentiable everywhere in 0 , ∞