Minimize[{If[x > 0, 3 x, x^2 - x - 4]}, x]

Minimize[{If[x > 0, 3 x, x^2 - x - a], -2 < a < 2}, x]

If a < 0 min = 0, If a > 0 min = - a

Minimize[{If[x > 0, 3 x, x^2 - x - a], -2 < a < 2}, x]

Minimize[{If[x > 0, 3 x, x^2 - x - a], -2 < a < 2}, x]

$Version

(* "10.0 for Mac OS X x86 (64-bit) (December 4, 2014)" *)

Clear[f]

f[x_, a_] = PiecewiseExpand@If[x > 0, 3 x, x^2 - x - a]

(* Piecewise[{{3*x, x > 0}}, -a - x + x^2] *)

Plot3D[f[x, a], {x, -3, 3}, {a, -2, 2}]

Minimize[{f[x, a], -2 < a < 2}, {x, a}]

(* Minimize::wksol: Warning: there is no minimum in the region in which the
objective function is defined and the constraints are satisfied; returning a
result on the boundary. >>

{-2, {x -> 0, a -> 2}} *)