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JEE Main2014MathematicsApplication of DerivativesActual

If non-zero real numbers b and c are such that m i n f x > m a x g x , where f x = x 2 + 2 b x + 2 c 2 and g x = - x 2 - 2 c x + b 2 , x ∈ R ; then c b lies in the interval

Options

  1. A2 , ∞
  2. B[ 1 2 , 1 2 )
  3. C0 , 1 2
  4. D1 2 , 2

Correct answer

A. 2 , ∞

Step-by-step solution

Given f x = x 2 + 2 b x + 2 c 2 and g x = - x 2 - 2 c x + b 2 For finding the minimum value of f ( x ) , f ' x = 2 x + 2 b = 0 ⇒ x = - b Also, f " x = 2 ⇒ f " - b = 2 > 0 So, f x has minimum value at x = - b Hence, f m i n = b 2 - 2 b 2 + 2 c 2 ⇒ f m i n = 2 c 2 - b 2 Similarly, for finding the maximum value of g ( x ) g ' x = - 2 x - 2 c = 0 ⇒ x = - c Also, g " x = - 2 g " - c = - 2 < 0 So, g x has maximum value at x = - c . Hence, g max = - c 2 + 2 c 2 + b 2

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