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Question:

For x ∈ (0, 5π/4), define f(x) = ∫₀ˣ √t - sin t dt. Then f(x) has

local minimum at π and 2π

local minimum at π and local maximum at 2π

local maximum at π and 2π

local maximum at π and local minimum at 2π

Solution:

f(x) = ∫₀ˣ √t - sin t dt
f'(x) = √x - sin x
Given x ∈ (0, 5π/4)
f'(x) changes sign from +ve to -ve at π
f'(x) changes sign from -ve to +ve at 2π
f(x) has local maximum at π and local minima at 2π
Hence, option 'D' is correct.