The following problem establish some more interesting results related to Newton divided differences:

i) Show thatf[x0?,x1?]=?01?f?((1?t)x0?+tx1?)dt

ii) Show that this can be used to justify the expression the expressionf[x,x]=f?(x)

Hint: Start with the integral expression and show that you get the divided difference on the left

Show that few] = [0 f'((1 t)xo +tm1) dt (5) Show that this can be used to justify the expression f [sax] = f'(:z:). Hint: Start with the integral expression and show that you get the divided difference on the left