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AP EAMCET202318 May 2023Morning ShiftMathematicsContinuity and DifferentiabilityActual

If f(x)= cases 1 2 (b^2-a^2 ), & 0 x a 1 2 b^2- x^2 6 - a^3 3 x , & a b cases

Options

  1. Af^ (a)=2 b
  2. Bf ^ ( a )=1
  3. Cf^ (a)=b^2-a^2
  4. Df ^ ( x ) is not differentiable at x = a

Correct answer

D. f ^ ( x ) is not differentiable at x = a

Step-by-step solution

L.H.D = _ h 0 f(a-h)-f(a) -h = _ h 0 1 2 (b^2-a^2 )- 1 2 (b^2-a^2 ) -h =0 Now, R.H.D. = _ h 0 f(a+h)-f(a) h aligned & = _ h 0 1 2 b^2- (a+h)^2 6 - a^3 3(a+h) - 1 2 (b^2-a^2 ) h & = _ h 0 1 h [ 1 2 a^2- (a+h)^3+2 a^3 6(a+h)^2 ] as h 0 aligned R.H.D. at x = a does not exist. f(x) is not differentiable at x=a . f ^ ( x ) is not differentiable at x = a .

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