Let f: S Y be a differentiable function on an open convex set S X. Then for every x1, x2 S, where A = {y

Let f: S †’ Y be a differentiable function on an open convex set S Š† X. Then for every x1, x2 ˆˆ S,

where
A = {y ˆˆ Y: y = Df[] (x1 - x2) for some ˆˆ [x1, x2]}.
Exercise 4.45 is illustrated in figure 4.8. The affine functions f(x1) + Df[x1](x2 - x1) and f(x1) + Df[x2](x2 - x1) bound the values of f(x) between x1 and x2.

Figure 4.8 The mean value inclusion theorem