Rolle's theorem

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If a real-valued function Template:Mvar is continuous on a closed interval Template:Closed-closed, differentiable on the open interval Template:Open-open, and f (a) = f (b), then there exists a Template:Mvar in the open interval Template:Open-open such that f ′(c) = 0.

In real analysis, a branch of mathematics, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Such a point is known as a stationary point. It is a point at which the first derivative of the function is zero. The theorem is named after Michel Rolle.

Standard version of the theorem

If a real-valued function Template:Mvar is continuous on a proper closed interval Template:Closed-closed, differentiable on the open interval Template:Open-open, and f (a) = f (b), then there exists at least one Template:Mvar in the open interval Template:Open-open such that $$f^{\prime }(c)=0.$$

This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.

History

Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.

Examples

Differentiability is not needed at the endpoints: Half circle

File:Semicircle.svg

A semicircle of radius Template:Mvar

For a radius r > 0, consider the function $$f(x)=\sqrt{r^2-x^2},x\in [-r,r].$$

Its graph is the upper semicircle centered at the origin. This function is continuous on the closed interval Template:Closed-closed and differentiable in the open interval Template:Open-open, but not differentiable at the endpoints −r and Template:Mvar. Since f (−r) = f (r), Rolle's theorem applies, and indeed, there is a point where the derivative of Template:Mvar is zero. The theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.

Differentiability is needed within the open interval: Absolute value

File:Absolute value.svg

The graph of the absolute value function

If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function $$f(x)=|x|,x\in [-1,1].$$

Then f (−1) = f (1), but there is no Template:Mvar between −1 and 1 for which the f ′(c) is zero. This is because that function, although continuous, is not differentiable at x = 0. The derivative of Template:Mvar changes its sign at x = 0, but without attaining the value 0. The theorem cannot be applied to this function because it does not satisfy the condition that the function must be differentiable for every Template:Mvar in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, Template:Mvar will still have a critical number in the open interval Template:Open-open, but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).

Functions with zero derivative

Rolle's theorem implies that a differentiable function whose derivative is Template:Tmath in an interval is constant in this interval.

Indeed, if Template:Mvar and Template:Mvar are two points in an interval where a function Template:Mvar is differentiable, then the function $$g(x)=f(x)-f(a)-\frac{f(b)-f(a)}{b-a}(x-a)$$ satisfies the hypotheses of Rolle's theorem on the interval Template:Tmath.

If the derivative of Template:Tmath is zero everywhere, the derivative of Template:Tmath is $$g^{\prime }(x)=-\frac{f(b)-f(a)}{b-a},$$ and Rolle's theorem implies that there is Template:Tmath such that $$0=g^{\prime }(c)=-\frac{f(b)-f(a)}{b-a}.$$

Hence, Template:Tmath for every Template:Tmath and Template:Tmath, and the function Template:Tmath is constant.

Generalization

The second example illustrates the following generalization of Rolle's theorem:

Consider a real-valued, continuous function Template:Mvar on a closed interval Template:Closed-closed with f (a) = f (b). If for every Template:Mvar in the open interval Template:Open-open the right-hand limit $$f^{\prime }(x^{+}):=\underset{h\to 0^{+}}{\lim }\frac{f(x+h)-f(x)}{h}$$ and the left-hand limit $$f^{\prime }(x^{-}):=\underset{h\to 0^{-}}{\lim }\frac{f(x+h)-f(x)}{h}$$

exist in the extended real line Template:Closed-closed, then there is some number Template:Mvar in the open interval Template:Open-open such that one of the two limits $$f^{\prime }(c^{+})\text{and}{f}^{\prime }(c^{-})$$ is ≥ 0 and the other one is ≤ 0 (in the extended real line). If the right- and left-hand limits agree for every Template:Mvar, then they agree in particular for Template:Mvar, hence the derivative of Template:Mvar exists at Template:Mvar and is equal to zero.

Remarks

• If Template:Mvar is convex or concave, then the right- and left-hand derivatives exist at every inner point, hence the above limits exist and are real numbers.
• This generalized version of the theorem is sufficient to prove convexity when the one-sided derivatives are monotonically increasing: $$f^{\prime }(x^{-})\le f^{\prime }(x^{+})\le f^{\prime }(y^{-}),x<y.$$

Proof of the generalized version

Since the proof for the standard version of Rolle's theorem and the generalization are very similar, we prove the generalization.

The idea of the proof is to argue that if f (a) = f (b), then Template:Mvar must attain either a maximum or a minimum somewhere between Template:Mvar and Template:Mvar, say at Template:Mvar, and the function must change from increasing to decreasing (or the other way around) at Template:Mvar. In particular, if the derivative exists, it must be zero at Template:Mvar.

By assumption, Template:Mvar is continuous on Template:Closed-closed, and by the extreme value theorem attains both its maximum and its minimum in Template:Closed-closed. If these are both attained at the endpoints of Template:Closed-closed, then Template:Mvar is constant on Template:Closed-closed and so the derivative of Template:Mvar is zero at every point in Template:Open-open.

Suppose then that the maximum is obtained at an interior point Template:Mvar of Template:Open-open (the argument for the minimum is very similar, just consider −f ). We shall examine the above right- and left-hand limits separately.

For a real Template:Mvar such that c + h is in Template:Closed-closed, the value f (c + h) is smaller or equal to f (c) because Template:Mvar attains its maximum at Template:Mvar. Therefore, for every h > 0, $$\frac{f(c+h)-f(c)}{h}\le 0,$$ hence $$f^{\prime }(c^{+}):=\underset{h\to 0^{+}}{\lim }\frac{f(c+h)-f(c)}{h}\le 0,$$ where the limit exists by assumption, it may be minus infinity.

Similarly, for every h < 0, the inequality turns around because the denominator is now negative and we get $$\frac{f(c+h)-f(c)}{h}\ge 0,$$ hence $$f^{\prime }(c^{-}):=\underset{h\to 0^{-}}{\lim }\frac{f(c+h)-f(c)}{h}\ge 0,$$ where the limit might be plus infinity.

Finally, when the above right- and left-hand limits agree (in particular when Template:Mvar is differentiable), then the derivative of Template:Mvar at Template:Mvar must be zero.

(Alternatively, we can apply Fermat's stationary point theorem directly.)

Generalization to higher derivatives

We can also generalize Rolle's theorem by requiring that Template:Mvar has more points with equal values and greater regularity. Specifically, suppose that

• the function Template:Mvar is n − 1 times continuously differentiable on the closed interval Template:Closed-closed and the Template:Mvarth derivative exists on the open interval Template:Open-open, and
• there are Template:Mvar intervals given by a₁ < b₁ ≤ a₂ < b₂ ≤ ⋯ ≤ a_n < b_n in Template:Closed-closed such that f (a_k) = f (b_k) for every Template:Mvar from 1 to Template:Mvar.

Then there is a number Template:Mvar in Template:Open-open such that the Template:Mvarth derivative of Template:Mvar at Template:Mvar is zero.

File:Rolle Generale.svg

The red curve is the graph of function with 3 roots in the interval Template:Closed-closed. Thus its second derivative (graphed in green) also has a root in the same interval.

The requirements concerning the Template:Mvarth derivative of Template:Mvar can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above with f Template:Isup in place of Template:Mvar.

Particularly, this version of the theorem asserts that if a function differentiable enough times has Template:Mvar roots (so they have the same value, that is 0), then there is an internal point where f Template:Isup vanishes.

Proof

The proof uses mathematical induction. The case n = 1 is simply the standard version of Rolle's theorem. For n > 1, take as the induction hypothesis that the generalization is true for n − 1. We want to prove it for Template:Mvar. Assume the function Template:Mvar satisfies the hypotheses of the theorem. By the standard version of Rolle's theorem, for every integer Template:Mvar from 1 to Template:Mvar, there exists a Template:Mvar in the open interval Template:Open-open such that f ′(c_k) = 0. Hence, the first derivative satisfies the assumptions on the n − 1 closed intervals [c₁, c₂], …, [c_{n − 1}, c_n]. By the induction hypothesis, there is a Template:Mvar such that the (n − 1)st derivative of f ′ at Template:Mvar is zero.

Generalizations to other fields

Rolle's theorem is a property of differentiable functions over the real numbers, which are an ordered field. As such, it does not generalize to other fields, but the following corollary does: if a real polynomial factors (has all of its roots) over the real numbers, then its derivative does as well. One may call this property of a field Rolle's property. More general fields do not always have differentiable functions, but they do always have polynomials, which can be symbolically differentiated. Similarly, more general fields may not have an order, but one has a notion of a root of a polynomial lying in a field.

Thus Rolle's theorem shows that the real numbers have Rolle's property. Any algebraically closed field such as the complex numbers has Rolle's property. However, the rational numbers do notTemplate:Snd for example, x³ − x = x(x − 1)(x + 1) factors over the rationals, but its derivative, $$3x^2-1=3(x-\frac{1}{\sqrt{3}})(x+\frac{1}{\sqrt{3}}),$$ does not. The question of which fields satisfy Rolle's property was raised in . For finite fields, the answer is that only F² and F⁴ have Rolle's property.

For a complex version, see Voorhoeve index.

See also

• Mean value theorem
• Intermediate value theorem
• Linear interpolation
• Gauss–Lucas theorem

External links

• Template:Springer
• Rolle's and Mean Value Theorems at cut-the-knot.
• Mizar system proof: http://mizar.org/version/current/html/rolle.html#T2