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MHT CET2011MathematicsContinuity and Differentiability

At the point x=1 , the function f(x)= array l x³-1,1 < x < x-1,- < x 1 array . is

Options

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

Correct answer

B. continuous and not differentiable

Step-by-step solution

LHL = _ x 1⁻ f(x)= _ x 1 (x-1)=0 aligned RHL &= _ x 1⁺ f(x) &= _ x 1 (x³-1 )=0 Also, f(1) &=1-1=0 aligned f is continuous at x=1 Now, L f^ (1)= _ h 0 f(1-h)-f(1) -h aligned &= _ h 0 (1-h)-1-0 -h &= _ h 0 -h -h =1 and R f^ (1) &= _ h 0 f(1+h)-f(1) h aligned array l = _ h 0 (1+h)³-1-0 h = _ h 0 1+h³+3 h+3 h²-1 h = _ h 0 h²+3+3 h=3 array f(x) is not differentiable at x=1

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