Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Black Book Advanced Problems in MathematicsMathematicsContinuity and Differentiability

If f(x) be a differentiable function satisfying f(y) f ( x y ) = f(x) x, y R, y 0 and f(1) 0 , f'(1) = 3 , then :

Options

  1. Asgn (f(x)) is non-differentiable at exactly one point
  2. B_ x 0 x^2( x - 1) f(x) = 0
  3. Cf(x) = x has 3 solutions
  4. Df(f(x)) - f^3(x) = 0 has infinitely many solutions

Correct answer

A. sgn (f(x)) is non-differentiable at exactly one point

Step-by-step solution

Open this question in the Quantrex Academy app for the full interactive solution, figures and similar PYQs.

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 Black Book Advanced Problems in Mathematics PYQs