AP EAMCET201825 Apr 2018Morning ShiftMathematicsContinuity and DifferentiabilityActual
Let f be defined on D = ℝ - - 1 , 1 by f x = | x | 1 - | x | , then
Options
- Af is differentiable on D
- Bf is differentiable on D except at x = 0
- Cf is continuous but not differentiable on D
- Df is differentiable but not continuous on D
Correct answer
B. f is differentiable on D except at x = 0
Step-by-step solution
Given: f ( x ) = | x | 1 - | x | Now, LHL = lim x → 0 - f x = lim h → 0 f 0 - h = lim h → 0 0 - h 1 - 0 - h = 0 RHL = lim x → 0 + f x = lim h → 0 f 0 + h = lim h → 0 0 + h 1 - 0 + h = 0 lim x → 0 - f x = lim x → 0 + f x = f 0 Hence, f x is continuous at x = 0 . Using differentiation: f x = - x 1 + x ,       x < 0 x 1 - x ,     x > 0 f ' x = - 1 1 + x + x 1 + x 2 ;       x < 0 1 1 + x + x 1 - x 2 ;     &