Most Important Selected Qs for JEE AdvancedMathematicsContinuity and Differentiability
If g(x)= array cc [f(x)], & x (0, 2 ) ( 2 , ) 3, & x= / 2 array . where [x] denotes the greatest integer function and f(x)= 2 ( x- ^n x )+ | x- ^n x | 2 ( x- ^n x )- | x- ^n x | , n R- 1 , then
Options
- Ag(x) is continuous and differentiable at x= / 2 , when 0 n 1
- Bg(x) is continuous and differentiable at x= / 2 , when n 1
- Cg ( x ) is continuous but not differentiable at x = / 2 , when 0 n 1
- Dg ( x ) is neither continuous nor differentiable, at x = / 2 , when n 1
Correct answer
B. g(x) is continuous and differentiable at x= / 2 , when n 1
Step-by-step solution
For 0 n 1, x ^ n x and for n 1, x ^ n x Now, for 0 n 1 , f(x)= 2 ( x- ^n x )- ( x- ^n x ) 2 ( x- ^n x )+ ( x- ^n x ) = 1 3 and for , f(x)= 2 ( x- ^n x )+ ( x- ^n x ) 2 ( x- ^n x )- ( x- ^n x ) =3 For n 1, g(x)=3, x (0, ) g ( x ) is continuous and differentiable at x = 2 , for 0 n 1 , g(x)= array cc [ 1 3 ]=0, & x (0, 2 ) ( 2 , ) 3, & x= 2 array . g ( x ) is not continuous at x = 2 , Hence, g ( x ) is also not differentiable at x = 2