To find the maximum value, we first find the derivative f'(x) = cos x - sin x. Setting f'(x) = 0 gives cos x = sin x, which implies tan x = 1, so x = π/4 within the given interval. The second derivative f''(x) = -sin x - cos x is negative at x = π/4 (f''(π/4) = -1/√2 - 1/√2 = -√2 < 0), confirming a local maximum. The value at x = π/4 is f(π/4) = sin(π/4) + cos(π/4) = 1/√2 + 1/√2 = 2/√2 = √2. Comparing this with the boundary values f(0) = 1 and f(π/2) = 1, the absolute maximum is √2. Option A is incorrect because 1 is the value at the boundaries, not the peak. Option C and D are incorrect based on the calculated trigonometric sum.