We begin with a commonsense geometrical fact:
somewhere between two zeros of a nonconstant continuous function f, the function must change direction
Rolle's Theorem Let f be differentiable on (a,b) and continuous on [a,b]. If f(a) = f(b) = 0, then there is at least one point c in (a,b) for which f¢(c) = 0.
Notice that both conditions on f are necessary. Without either one,
the statement is false!
Though the theorem seems logical, we cannot be sure that it is always true without a proof.
The Mean Value Theorem is a generalization of Rolle's Theorem: We now let f(a) and f(b) have values other than 0 and look at the secant line through (a, f(a)) and (b, f(b)). We expect that somewhere between a and b there is a point c where the tangent is parallel to this secant.
That is, the slopes of these two lines are equal. This is formalized in the Mean Value Theorem. Mean Value Theorem Let f be differentiable on (a,b) and continuous on [a,b]. Then there is at least one point c in (a,b) for which
Here, f¢(c) is the slope of the tangent at c, while (f(b)f(a))/( ba) is the slope of the secant through a and b. Intuitively, we see that if we translate the secant line in the figure upwards, it will eventually just touch the curve at the single point c and will be tangent at c. However, basing conclusions on a single example can be disastrous, so we need a proof.
Consequences of the Mean Value Theorem The Mean Value Theorem is behind many of the important results in calculus. The following statements, in which we assume f is differentiable on an open interval I, are consequences of the Mean Value Theorem:
Mean Value Theorem Let f be differentiable on (a,b) and continuous on [a,b]. Then there is at least one point c in (a,b) for which
