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AP EAMCET201923 Apr 2019Morning ShiftMathematicsContinuity and DifferentiabilityActual

If a function (f ) is defined by : ( aligned f(x) & =0, when x=1, & =x^3-1, when 1 < x < , aligned ) (=x-1 ), when (- < x < 1 ), then at (x=1, f ) is

Options

  1. Acontinuous and differentiable
  2. Bcontinuous but not differentiable
  3. Cdiscontinuous and differentiable
  4. Ddiscontinuous and not differentiable

Correct answer

B. continuous but not differentiable

Step-by-step solution

We have, (f(x)= array rc x-1, & - < x < 1 0, & x=1 x^3-1, & 1 < x < array . ) Now, (LHL at (x=1 ) ) ( aligned & = _ x 1 (x-1) & = l - l = 0 aligned ) (RHL at (x=1 ) ) (= _ x 1 x^3-1 ) (=( l )^3- 1 =0 and f( l )=0 ) ( LHL = RHL =f( l ) ) So, (f(x) ) is continuous at (x=1 ) Now, (LHD at (x=1 ) ) (= _ x 1 (x-1)-0 x-1 =1 ) and (RHD at (x=1 ) ) ( aligned & = _ x 1 (x^3-1 )-0 x-1 & = _ x 1 (x^2+x+1 )=3 aligned ) ( ) LHD ( ) RHD So, (f(x) ) is not differentiable at (x=1 )

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