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The mean value theorem is a generalization of Rolle's theorem, which assumes () = (), so that the right-hand side above is zero. The mean value theorem is still valid in a slightly more general setting.
In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. [1] Statement of the theorem
In mathematics, the mean value problem was posed by Stephen Smale in 1981. [1] This problem is still open in full generality. The problem asks: For a given complex polynomial of degree [2] A and a complex number , is there a critical point of (i.e. ′ =) such that
Moreover, , =, = = for all 0 < s < r so that Δu = 0 in Ω by the fundamental theorem of the calculus of variations, proving the equivalence between harmonicity and mean-value property. This statement of the mean value property can be generalized as follows: If h is any spherically symmetric function supported in B(x, r) such that =, then () = ().
These refinements of Taylor's theorem are usually proved using the mean value theorem, whence the name. Additionally, notice that this is precisely the mean value theorem when k = 0 {\textstyle k=0} .
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. [1]
The point () is called the mean value of () on [,]. So we write f ¯ = f ( c ) {\displaystyle {\bar {f}}=f(c)} and rearrange the preceding equation to get the above definition. In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by
In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers. It is an important inequality in analytic number theory , named for I. M. Vinogradov . More specifically, let J s , k ( X ) {\displaystyle J_{s,k}(X)} count the number of solutions to the system of k {\displaystyle k} simultaneous Diophantine ...