JEE Main2014MathematicsContinuity and DifferentiabilityActual

Let f(x)=x|x|, g(x)= x and h(x)=(g f)(x) . Then

Options

Ah ( x ) is not differentiable at x =0 .
Bh ( x ) is differentiable at x =0 , but h ^ ( x ) is not continuous at x=0
Ch ^ ( x ) is continuous at x =0 but it is not differentiable at x=0
Dh ^ ( x ) is differentiable at x =0

Correct answer

C. h ^ ( x ) is continuous at x =0 but it is not differentiable at x=0

Step-by-step solution

Let f(x)=x|x|=x|x|, g(x)= x and h(x)=g o f(x)=g[f(x)] aligned & h(x)= cases x^2, & x 0 - x^2, & x < 0 cases & Now, h^ (x)= array cc 2 x x^2, & x 0 -2 x x^2, & x < 0 array . aligned Since, L.H.L and R.H.L at x=0 of h^ (x) is equal to 0 therefore h^ (x) is continuous at x=0 Now, suppose h^ (x) is differentiable aligned & h^ (x) &= cases 2 ( x^2+2 x^2 (- x^2 ), . & x 0 2 (- x^2+2 x^2 x^2 ), & x < 0 cases aligned Since, L.H.L and R.H.L at x=0 of h^ (x) are different therefore h^ (x) is not continuous. h^ (x) is not dif

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