Rolle's theorem

If the real function Template:Mvar is continuous on the proper closed interval Template:Closed-closed, differentiable on the open interval Template:Open-open, and Template:Math, then there exists Template:Mvar in Template:Open-open such that Template:Math.

In calculus and real analysis, Rolle's theorem (or lemma) states that a real-valued differentiable function which attains equal values at two distinct points must have a stationary point somewhere between them, that is, a point where its derivative is zero. The theorem is named after Michel Rolle.

The theorem is a special case of, and is used to prove, the mean value theorem.

Statement

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

History

In the 12th century Bhāskara II states an early form of Rolle's theorem in his work, though it was stated without a modern formal proof. 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 and counterexamples

Differentiability is not needed at the endpoints: Half circle

For a positive real number Template:Math, consider the function $f:[-r,r]\to ℝ$ such that $f(x)=\sqrt{r^2-x^2}$ for all $x\in [-r,r].$

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 Template:Math and Template:Mvar, as the graph of Template:Mvar has vertical tangents at those points. Since Template:Math, 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

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 Template:Math, but there is no Template:Mvar between −1 and 1 for which the Template:Math is zero. This is because that function, although continuous, is not differentiable at Template:Math. The derivative of Template:Mvar changes its sign at Template:Math, 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.

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.

Requirement for completeness of domain

Rolle's theorem relies on the completeness of the real numbers. If the domain is a non-complete densely ordered field, one can exploit the 'holes' in the domain to construct a function that does not obey Rolle's theorem (assuming that the definitions of continuous and differentiable are adjusted for the incomplete ordered domain).

For example, consider $f:\mathbb{Q}\to \mathbb{Q}$, such that for all $x\in \mathbb{Q}$, $f(x)=(x^2-2{)}^2$. Then $f$ is continuous on $[0,2]\cap \mathbb{Q}$, differentiable on $(0,2)\cap \mathbb{Q}$, $f(0)=4$ and $f(2)=4$, but there exists no $x\in (0,2)\cap \mathbb{Q}$ such that $f^{\prime }(x)=0$. In the real numbers, this value would be $\sqrt{2}$, but $\sqrt{2}$ is not in $(0,2)\cap \mathbb{Q}$.

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 Template:Math. 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 Template:Math and the other one is Template:Math (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 Template:Math, 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 Template:Math). We shall examine the above right- and left-hand limits separately.

For a real Template:Mvar such that Template:Math is in Template:Closed-closed, the value Template:Math is smaller or equal to Template:Math because Template:Mvar attains its maximum at Template:Mvar. Therefore, for every Template:Math, $$\frac{f(c+h)-f(c)}{h}\le 0,$$ 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 Template:Math, 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 Template:Math 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 Template:Math in Template:Closed-closed such that Template:Math 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.

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 Template:Math 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 Template:Math vanishes.

Proof

The proof uses mathematical induction. The case Template:Math is simply the standard version of Rolle's theorem. For Template:Math, take as the induction hypothesis that the generalization is true for Template:Math. 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 Template:Math. Hence, the first derivative satisfies the assumptions on the Template:Math closed intervals Template:Math. By the induction hypothesis, there is a Template:Mvar such that the Template:Mathst derivative of Template:Math 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, Template:Math 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 Template:Harvnb. For finite fields, the answer is that only Template:Math and Template:Math have Rolle's property.

For a complex version, see Voorhoeve index.

See also

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

Further reading

• Template:Cite book
• Template:Cite book

External links

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