For a real valued function of one variable, f(x); if P and Q are two points on the graph of said function, and these points are equidistant from the xaxis (ie; have the same yvalue); then, there is a point, R, on the graph, between P and Q; such that, the tangent to the graph ( through the point R) is both horizontal, and parallel, to the xaxis.
Or symbolically:
Or:

Example Consider the graph of x^{2}...For any two points (on this graph), having equivalent yvalues, there is the point (0,0), where the slope is zero.
Note on Diferentiability Should one remove the clause that a function be differentiable, upon the open interval ]a,b[; then, f(x) will have a critical point at R; although, it might not have a horizontal tangent.
It states that, for smooth curves, if the function is equal at two points there must be a stationary point somewhere between them. All the assumptions are necessary. For example, if f(x) = x, the absolute value of x, then we have that f(1) = f(1), but there is no x between 1 and 1 for which f '(x) = 0.
Rolle's Theorem is used in proving the mean value theorem, which can be seen as a generalisation of it.
Proof of Rolle's Theorem: The idea of the proof is to argue that if f(a) = f(b) then f must attain either a maximum or a minimum somewhere between a and b, and f ' (x) = 0 at either of these points.
Now, by assumption, f is continuous on [a , b], and by the continuity property is bounded and attains both its maximum and its minimum at points of [a , b]. If these are both attained at endpoints of [a , b] then f is constant on [a , b] and so f ' (x) = 0 at every point of (a , b).
Suppose then that the maximum is obtained at an interior point x of (a , b) ( the argument for the minimum is very similar). We wish to show that f ' (x) = 0. We shall examine the lefthand and righthand derivatives separately.
For y just below x, ( f(x)  f(y) ) / (x  y) is nonnegative, since x is a maximum. Thus the limit lim_{y>x} is nonnegative. (Note that we assume that f is differentiable to guarantee that the lefthand and righthand derivatives exist; it does not follow from the other assumptions).
For y just greater than x, ( f(x)  f(y) ) / (x  y) is nonpositive. Thus lim_{y>x+} is nonpositive.
Finally, since f is differentiable at x, these two limits must be equal and hence are both 0. This implies that f ' (x) = 0.
Generalization: The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting: We only need to assume that f : [a , b] > R is continuous on [a , b], that f(a) = f(b), and that for every x in (a , b) the limit lim_{h>0} (f(x+h)f(x))/h exists or is equal to +/ infinity.
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