NTA Abhyas JEE Main2020MathematicsApplication of DerivativesPractice

The maximum value of f x = sin ⁡ 2 x sin ⁡ x + cos ⁡ x in the interval 0 , π 2 is

Options

A2
B1 2
C1
D1 2

Correct answer

B. 1 2

Step-by-step solution

f x = sin ⁡ x + cos ⁡ x 2 - 1 sin ⁡ x + cos ⁡ x Substituting sin ⁡ x + cos ⁡ x = t , the expression becomes g t = t 2 - 1 t = t - 1 t Now, t = sin ⁡ x + cos ⁡ x ,   ∀ x ∈ 0 , π 2 t ∈ 1 , 2 g ' t = 1 + 1 t 2 > 0 Therefore, g t is an increasing function. Maximum value = g 2 = 2 - 1 2 = 1 2

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