Differentiating f(x), we get f'(x) = cos x - sin x. Setting f'(x) = 0 gives cos x = sin x, or tan x = 1, which means x = π/4 in the given interval. The value at x = π/4 is sin(π/4) + cos(π/4) = 1/√2 + 1/√2 = 2/√2 = √2. Checking endpoints: f(0) = 1 and f(π/2) = 1. Since √2 ≈ 1.414 > 1, the maximum value is √2.