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COMEDK2024Morning ShiftMathematicsApplication of DerivativesActual

If f(x)=2 x^3+9 x^2+ x+20 is a decreasing function of x in the largest possible interval (-2,-1) , then is equal to

Options

  1. A12
  2. B-12
  3. C-6
  4. D6

Correct answer

A. 12

Step-by-step solution

Given f(x) = 2x^3 + 9x^2 + x + 20 . The derivative is f'(x) = 6x^2 + 18x + . For f(x) to be a decreasing function, f'(x) 0 for all x in the interval (-2, -1) . The quadratic f'(x) = 6x^2 + 18x + represents a parabola opening upwards. For it to be non-positive in the interval (-2, -1) , the roots of f'(x) = 0 must be -2 and -1 . If the roots are -2 and -1 , then f'(x) = 6(x + 2)(x + 1) = 6(x^2 + 3x + 2) = 6x^2 + 18x + 12 . Comparing 6x^2 + 18x + 12 with 6x^2 + 18x + , we get = 12 . Checking the condition, f'(x) = 6(

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