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AP EAMCET20224 Jul 2022Evening ShiftMathematicsContinuity and DifferentiabilityActual

If f x = 1 x ; x ≥ 1 a x 2 + b ; – 1 < x < 1 is differentiable ∀ x , then one of the value of a and b is-

Options

  1. Aa = 1 2 , b = – 3 2
  2. Ba = – 1 2 , b = 3 2
  3. Ca = 3 2 , b =   1 2
  4. DNone of these

Correct answer

B. a = – 1 2 , b = 3 2

Step-by-step solution

Given, f x   = 1 x ; x   ≥ 1 a x 2 + b ; – 1 < x < 1 ⇒ f x = - 1 x x ≤ - 1 a x 2 + b - 1 < x < 1 1 x x > 1 Also given funciton at x = 1 is continuous So, a . 1 + b = 1   . . . . . . 1 Also at x = 1 differentiable 2 a x = – 1 x 2 Now at x = 1 ⇒ 2 a = – 1 So, a = – 1 2 Now using equation 1 we get, b = 3 2 . 

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