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BITSAT2019MathematicsApplication of DerivativesActual

The least value of the function f ( x ) = ∫ 0 x 3 sin x + 4 cos x d x in the interval 5 π 4 , 4 π 3 is

Options

  1. A3 - 3 2
  2. B5 - 4 3 2
  3. C7 - 4 3 2
  4. D9 - 4 3 2

Correct answer

D. 9 - 4 3 2

Step-by-step solution

We have, f x = ∫ 0 x 3 sin x + 4 cos x d x f ' x = 3 sin x + 4 cos x = 5 sin x + tan - 1 4 3 5 π 4 < x < 4 π 3 π < x + tan - 1 4 / 3 < 2 π ∵   f ' x < 0 , i.e. f x is decreasing. ∵ least value of f x is 5 π 4 , 4 π 3 = f 4 π 3 least f x = ∫ 0 4 π / 3 3 sin x + 4 cos x d x = - 3 cos x + 4 sin x 0 4 π / 3 = - 3 - 1 2 - 1 + 4 - 3 2 = 9 - 4 3 2

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