A differentiable function f(x) has a local maximum at x = 0. Let y = 2f(x) + ax - b. The function y has a relative maxima at x = 0 for

y = 2f(x) + ax - b. Then y' = 2f'(x) + a. At x = 0, y'(0) = 2f'(0) + a = 0 + a = a. For x = 0 to be a critical point, we need a = 0. Then y'' = 2f''(x), and y''(0) = 2f''(0) < 0, so x = 0 is a maximum. The value of b just shifts y vertically and doesn't affect the location of extrema. So y has relative maxima at x = 0 for all b and a = 0.