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AP EAMCET20224 Jul 2022Morning ShiftMathematicsApplication of DerivativesActual

The condition that f ( x ) = a x 3 + b x 2 + c x + d has no extreme value, is

Options

  1. Ab 2 > 3 a c
  2. Bb 2 = 4 a c
  3. Cb 2 = 3 a c
  4. Db 2 < 3 a c

Correct answer

D. b 2 < 3 a c

Step-by-step solution

Given curve is f ( x ) = a x 3 + b x 2 + c x + d On differentiating w.r.t. x , we get f ' ( x ) = 3 a x 2 + 2 b x + c For extremum, f ' ( x ) = 0 ⇒ 3 a x 2 + 2 b x + c = 0 Since, it has no extremum value. ∴ Discriminant < 0 ⇒ ( 2 b ) 2 - 4 × 3 a × c < 0 ⇒ 4 b 2 - 12 a c < 0 ⇒ b 2 - 3 a c < 0 ⇒ b 2 < 3 a c

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