Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202227 Jul 2022Evening ShiftMathematicsContinuity and DifferentiabilityActual

Let f x = 2 + x - x - 1 + x + 1 , x ∈ R . Consider S 1 : f ' - 3 2 + f ' - 1 2 + f ' 1 2 + f ' 3 2 = 2 S 2 : ∫ - 2 2 f x d x = 12 Then,

Options

  1. Aboth S 1 and S 2 are correct
  2. Bboth S 1 and S 2 are wrong
  3. Conly S 1 is correct
  4. Donly S 2 is correct

Correct answer

D. only S 2 is correct

Step-by-step solution

Given f x = 2 + x - x - 1 + x + 1 ,   x ∈ R ⇒ f x = - x , x < - 1 x + 2 , - 1 ≤ x < 0 3 x + 2 , 0 ≤ x < 1 x + 4 , x ≥ 1 ⇒ f ' x = - 1 , x < - 1 1 , - 1 < x < 0 3 , 0 < x < 1 1 , x > 1 ∴    f ' - 3 2 + f ' - 1 2 + f ' 1 2 + f ' 3 2 = - 1 + 1 + 3 + 1 = 4 i.e. S 1 is incorrect. Now ∫ - 2 2 f x d x = ∫ - 2 - 1 f x d x + ∫ - 1 0 f x d x + ∫ 0 1 f x d x + ∫ 1 2 f x d x = - x 2 2 -

Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

Let R denote the set of all real numbers. Let f : R R be an arbitrary function and let g : R R be the function defined by g(x) = x f(x) , for all x R . Then which of the following 2026Consider the function f : (- 2 , 2 ) (- , ) defined by f(x) = (|x| + |x - 1|) x + [x x] , where [x x] is the greatest integer less than or equal to x x . Let be the total number of 2026For a real number , let [ ] denote the greatest integer less than or equal to . For a finite set S , let |S| denote the number of elements in the set S . Consider the functions f : 2026For the function f(x) = e^ |x| - |x| , x R , consider the following statements: Statement I: f is differentiable for all x R . Statement II: f is increasing in (- , - 2 ) . In the 2026Let f(x) = cases 1 3 , & x /2 b(1- x) ( -2x)^2 , & x > /2 cases . If f is continuous at x= /2 , then the value of ₀^ 3b-6 |x^2+2x-3| ,dx is: 2026Let f(x) = cases x^3 + 8 ; & x Then the number of points, where the function g f is discontinuous, is __________. 2026Let f(x)= cases e^ x-1 , & x<0 x^2-5x+6, & x 0 cases and g(x)=f(|x|)+|f(x)| . If the number of points where g is not continuous and is not differentiable are and respectively, then 2026The number of points, at which the function f(x) = 6x, 2 + 3x^2 + |x - 1| | |x^2 - 1 4 | | , x (- , ) , is not differentiable, is _____. 2026 Full Continuity and Differentiability list All JEE Main PYQs