In general if y is bigger than x and we fix f(x), the larger f '(z) is between x and y the larger f(y) will be: (that which increases faster from the same start becomes bigger.)

This if we define a new fuction g(z) with g(x) = f(x) and g ' (z) f ' (z) then we will have g(y) f(y)

In particular, if we let g'(z) between x and y be the maximum value that f '(z) takes in this interval, we will have g(y) f(y), given g(x) = f(x).

But if g' is this constant in this interval, g is a straight line and we can compute g(y):

g(y) = g(x) + (y - z)g'(z)

this proves the bound:

f(y) f(x) + (y-z)Max _{x < z < y}[f '(z)]

All other statements are proven by exactly parallel arguments, applied either to higher derivatives or to min rather than max. The direction of the inequalities reverse for y < x.