JEE Main2012MathematicsContinuity and DifferentiabilityActual

If x+|y|=2 y , then y as a function of x , at x=0 is

Options

Adifferentiable but not continuous
Bcontinuous but not differentiable
Ccontinuous as well as differentiable
Dneither continuous nor differentiable

Correct answer

B. continuous but not differentiable

Step-by-step solution

Given x+|y|=2 y aligned & x+y=2 y or x-y=2 y & x=y or x=3 y aligned This represent a straight line which passes through origin. Hence, x+|y|=2 y is continuous at x=0 . Now, we check differentiability at x=0 gathered x+|y|=2 y x+y=2 y, y 0 x-y=2 y, y < 0 gathered Thus, f(x)= array ll x, & y < 0 x / 3, & y 0 array aligned & Now, L.H.D. = _ h 0⁻ f(x+h-) f x -h ( ) & = _ h 0⁻ x+h-x -h =-1 & R.H.D = _ h 0⁺ f(x+h-) f x h ( ) & = _ h 0⁺ x+h 3 - x 3 h = _ h 0⁺ 1 3 = 1 3 & aligned Since, L.H.D R.H.D. at x=0 given function i

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