Introduction to the Mathematical and Statistical Foundations of Econometrics
The Mean Value Theorem
Consider a differentiable real function f (x) displayed as the curved line in Figure II.1. We can always find a point c in the interval [a, b] such that the slope of f (x) at x = c, which is equal to the derivative f(c), is the same as the slope of the straight line connecting the points (a, f(a)) and (b, f (b)) simply by shifting the latter line parallel to the point where it becomes tangent to f (x). The slope of this straight line through the points (a, f (a)) and (b, f (b)) is (f (b) - f (a))/(b - a). Thus, at x = c we have f''(c) = (f (b) - f (a))/(b - a), or equivalently, f (b) = f (a) + (b - a)f(c). This easy result is called the mean value theorem. Because this point c can also be expressed as c = a + X(b - a), with 0 < X = (c - a)/(b - a) < 1, we can now state the mean value theorem as follows:
Theorem II.8(a): Let f (x) be a differentiable real function on an interval [a, b] with derivative f'(x). For any pair ofpoints x, x0 є [a, b] there exists a X є [0, 1] such that f (x) = f (x0) + (x - x0) f'(x0 + X(x - x0)).
This result carries over to real functions of more than one variable:
Theorem II.8(b): Let f (x) be a differentiable real function on a convex subset C of R*. For any pair of points x, x0 є C there exists a X є [0, 1] such that
f(x) = f(x0) + (x - x0)T(9/9yT)f(y)|y=x0+X(x-x0).
