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AP EAMCET202120 Aug 2021Evening ShiftMathematicsApplication of DerivativesActual

If g ( x ) = 1 6 f 3 x 2 - 1 + 1 2 f 1 - x 2 , ∀ x ∈ R , where f '' ( x ) > 0 , ∀ x ∈ R . Then g ( x ) is increasing in the interval

Options

  1. A- 1 2 , 0 ∪ 1 2 , ∞
  2. B- 1 2 , 1 2
  3. C( - 1 , 0 ) ∪ ( 1 , 2 )
  4. D- ∞ , - 1 2 ∪ 1 2 , ∞

Correct answer

A. - 1 2 , 0 ∪ 1 2 , ∞

Step-by-step solution

g x = 1 6 f 3 x 2 - 1 + 1 2 f 1 - x 2 ⇒ g ' x = 1 6 f ' 3 x 2 - 1 · 6 x + 1 2 f ' 1 - x 2 · - 2 x ⇒ g ' x = x f ' 3 x 2 - 1 - f ' 1 - x 2 For g ( x ) to be increasing function ⇒ g ' x = x f ' 3 x 2 - 1 - f ' 1 - x 2 > 0 So,if   x > 0 ⇒ f ' 3 x 2 - 1 - f ' 1 - x 2 > 0 ⇒ f ' 3 x 2 - 1 > f ' 1 - x 2 ⇒ 3 x 2 - 1 > 1 - x 2 ⇒ 4   x 2 > 2 ⇒ x 2   > 1 2 ⇒ x ∈ 1

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