Mean value theorem

In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to the instantaneous rate of change at some point in the interval. For example, if a car smoothly travels a certain distance over a given finite time interval, then at some moment during the trip, its instantaneous speed equals its average speed for the whole trip. The theorem states precisely that if a real-valued function is continuous on a closed interval [ a , b ] {\displaystyle [a,b]} , with a < b {\displaystyle a<b} , and differentiable on the interior ( a , b ) {\displaystyle (a,b)} , then there is at least one point in ( a , b ) {\displaystyle (a,b)} where the derivative equals the function's average rate of change over the whole interval. Geometrically, this means that at some point the tangent to the graph is parallel to the secant line through the interval's endpoints. It is used in proving other general properties of differentiable functions.

History A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (1380–1460), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823. Many variations of this theorem have been proved since then.

Statement

Let f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } be a continuous function on the closed interval [ a , b ] {\displaystyle [a,b]} , and differentiable on the open interval ( a , b ) {\displaystyle (a,b)} , where a < b {\displaystyle a<b} . Then there exists some c {\displaystyle c} in ( a , b ) {\displaystyle (a,b)} such that:

f ′ ( c ) = f ( b ) − f ( a ) b − a . {\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.}

The mean value theorem is a generalization of Rolle's theorem, which assumes f ( a ) = f ( b ) {\displaystyle f(a)=f(b)} , so that the right-hand side above is zero. The mean value theorem is still valid in a slightly more general setting. One only needs to assume that f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } is continuous on [ a , b ] {\displaystyle [a,b]} , and that for every x {\displaystyle x} in ( a , b ) {\displaystyle (a,b)} the limit

lim h → 0 f ( x + h ) − f ( x ) h {\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}

exists as a finite number or equals ∞ {\displaystyle \infty } or − ∞ {\displaystyle -\infty } . If finite, that limit equals f ′ ( x ) {\displaystyle f'(x)} . An example where this version of the theorem applies is given by the real-valued cube root function mapping x ↦ x 1 / 3 {\displaystyle x\mapsto x^{1/3}} , whose derivative tends to infinity at the origin.

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