f is in C[a,b] , a <= x_0 <= x_1 <= b ; and A is the operator that turns f into

interpolating polynomial of f with degree 1 at the interpolation points x_0 and x_1 ;

i.e.

Af(x) = {(f(x_1)-f(x_0)) / (x_1-x_0)}.x + (x_1.f(x_0) - x_0.f(x_1))

How can we show that A is positive if and only if x_0 = a , x_1 = b ?