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KVPY2011MathematicsApplication of Derivatives

Let f(x)=x³+a x²+b x+c , where a, b, c are real numbers. If f(x) has a local minimum at x=1 and a local maximum at x=- 1 3 and f(2)=0 , then _ -1 ¹ f(x) d x equals-

Options

  1. A14 3
  2. B-14 3
  3. C7 3
  4. D-7 3

Correct answer

B. -14 3

Step-by-step solution

f^ (x)=3 (x²- 2 3 x- 1 3 )=3 x²-2 x-1f(x)=x³-x²-x+ (2)=8-4-2+ =0 =-2f(x)=x³-x²-x-2 _ -1 ¹ f(x) d x=-2 ₀¹ (x²+2 ) d x=-2 ( 1 3 +2 )= -14 3

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