This function is a piecewise function. When x≥a, f (x) = x2+x-a-1= (x+1/2) 2-(a+5/4), and the parabolic opening is upward.

The symmetry axis is x=- 1/2, which is called function f 1(x).

When x

1) when a >; When 1/2 and f 1(x) are increasing function, f2(x) takes the minimum value at the symmetry axis x= 1/2, and f2( 1/2)=a-5/4.

At this time, the minimum value of the function f(x) is f2( 1/2), that is, fmin=a-5/4.

2) When

At this time, the minimum value of the function f(x) is f 1(- 1/2), that is, fmin=-(a+5/4).

3) when-1/2≤a≤ 1/2, f 1(x) is increasing function, f2(x) is a decreasing function, and f(x) takes the minimum value at x=a, and fmin = a 2-/kloc-0.

Therefore, the minimum value of f(x): fmin=-(a+5/4), a.

Fmin = a 2- 1，- 1/2≤a≤ 1/2