JEE Main20199 Jan 2019Morning ShiftMathematicsContinuity and DifferentiabilityActual
Let f : R → R be a function defined as f x = 5 , i f x ≤ 1 a + b x , i f 1 < x < 3 b + 5 x , i f 3 ≤ x < 5 30 , i f x ≥ 5 Then f is:
Options
- Acontinuous if a = - 5 and b = 10
- Bcontinuous if a = 0 and b = 5
- Cnot continuous for any values of a and b
- Dcontinuous if a = 5 and b = 5
Correct answer
C. not continuous for any values of a and b
Step-by-step solution
f x = 5 : x ≤ 1 a + b x : 1 < x < 3 b + 5 x : 3 ≤ x < 5 30 : x ≥ 5 For f ( x ) to be continuous at x = 5 l i m x → 5 - f x = l i m x → 5 + f x = f 5 ⇒ l i m x → 5 - b + 5 x = 30 ⇒ b + 25 = 30 ⇒ b = 5 For f ( x ) to be continuous at x = 1 l i m x → 1 + f x = l i m x → 1 - f x = f 1 ⇒ l i m x → 1 + a + 5 x = 5 ⇒ a = 0 Now at x = 3 , L H L = l i m x → 3 - 0 + 5 x = 15 R H L = l i m x → 3 + 5 + 5 x = 20 Hence function f ( x ) is discontinuous at x = 3 for all a , b ∈ R .