F'(x)>0, and f(x) increases monotonically.

Let a primitive function of f(x) be G(x)

F(x)=∫[0: 1]|f(x)-f(t)|dt

=∫[0:x][f(x)-f(t)]dt+∫[x: 1][f(t)-f(x)]dx

=[t f(x)-G(t)]|[0:x]+[G(t)-t f(x)]|[x: 1]

=[xf(x)-G(x)]-[0 f(x)-G(0)]+[G( 1)- 1 f(x)]-[G(x)-x f(x)]

= 2xf(x)-2G(x)-f(x)+G(0)+G( 1)

f '(x)=[2xf(x)-2G(x)-f(x)+G(0)+G( 1)]'

=2f(x)+2xf'(x)-2f(x)-f'(x)

=(2x- 1)f'(x)

Let F'(x)≥0 and get (2x- 1)f'(x)≥0.

F'(x)>0, so only 2x- 1≥0.

x≥?

F(x) in [0,? ] monotonously decreasing, in [? , 1] monotonically increasing.

The extreme point of the function is x=? This extreme value is the minimum value.