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AP EAMCET201825 Apr 2018Morning ShiftMathematicsTrigonometric Ratios & IdentitiesActual

The maximum and minimum values of the function f : ℝ → ℝ defined by f x = 5 cos x + 3 cos x + π 3 + 8 for all x ∈ ℝ , are respectively

Options

  1. A15 ,   1
  2. B8 ,   - 8
  3. C- 7 ,   - 15
  4. D1 ,   - 15

Correct answer

A. 15 ,   1

Step-by-step solution

Given, f ( x ) = 5 cos x + 3 cos x + π 3 + 8 = 5 cos x + 3 cos x cos π 3 - sin x sin π 3 + 8 = 5 cos x + 3 cos x 1 2 - sin x 3 2 + 8 = 5 cos x + 3 2 cos x - 3 3 2 sin x + 8 = 13 2 cos x - 3 3 2 sin x + 8 Now, we know that A sin x + B cos x ∈ - A 2 + B 2 , A 2 + B 2 Here, we have A = 13 2 and B = 3 3 2 Therefore, - 13 2 2 + 3 3 2 2 + 8 ≤ f x ≤ 13 2 2 + 3 3 2 2 + 8 ⇒ 1 ≤ f x ≤ 15 Minimum value of f x = 1 Maximum value of f x = 15

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