JEE Main2016MathematicsContinuity and DifferentiabilityActual

For x ∈ R , f x = log ⁡ 2 - sin ⁡ x and g x = f f x , then

Options

Ag ' 0 = - cos ⁡ log ⁡ 2
Bg is differentiable at x = 0 and g ' 0 = - sin ⁡ log ⁡ 2
Cg is not differentiable at x = 0
Dg ' 0 = cos ⁡ log ⁡ 2

Correct answer

D. g ' 0 = cos ⁡ log ⁡ 2

Step-by-step solution

In the neighborhood of x = 0 , f x = log ⁡ 2 - sin ⁡ x ∴ g x = f ( f x ) = log ⁡ 2 - sin ⁡ f x = log ⁡ 2 - sin log ⁡ 2 - sin ⁡ x ∴ g ′ x = - cos ⁡ log ⁡ 2 - sin ⁡ x – cos ⁡ x ∴ g ′ 0 = cos ⁡ log ⁡ 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 2026 Consider 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 2026 For 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 : 2026 For 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 2026 Let 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: 2026 Let f(x) = cases x^3 + 8 ; & x Then the number of points, where the function g f is discontinuous, is __________. 2026 Let 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 2026 The 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