Cauchy-Schwartz inequality is often used in mathematical analysis, and it is also widely used in competition mathematics and Schwartz advanced mathematics. Three proof methods are given below to deepen the understanding of this inequality and benefit teaching. Theorem (Cauchy-Schwartz inequality): If a 1, a2, …, an and b 1, b2, …, bn are arbitrary real numbers, then (NK = 1 ∑ akbk) 2 ≤ (NK =1∑ ak2).

Mathematically, Cauchy-Schwartz inequality, also known as Schwartz inequality or Cauchy-Ban Iacov-Schwartz inequality, is an inequality applied to many occasions, such as vectors in linear algebra, integrals of products of infinite series in mathematical analysis, variances and covariances in probability theory, etc. The inequalities are augustin louis cauchy, hermann amandus schwarz, and Victor Jakovljevic Ban Iacov (викторяовлев).

Cauchy-Schwartz inequality says that if x and y are elements of real or complex inner product space, then

& lt Mathematics & gt\ big | \ Lelang X, y \ Langle \ Big | 2 \ Leq \ Lelang X, x \ Langle \ cdot \ Lelang Y, y \ Langle & lt/math & gt；; .

This equation holds if and only if x and y are linearly related.

An important result of Cauchy-Schwartz inequality is that the inner product is a continuous function.

There is another form of Cauchy-Schwartz inequality, which can be expressed by norm:

& lt/math & gt；|\ Lelang x, y \ Rangele | \ Leq \ | x \ | \ CDOT \ | y \ | \,</Math > .

L'Hospital's rule

Under certain conditions, it is a method to determine the undetermined value by differentiating the numerator and denominator respectively, and then finding the limit.

set up

(1) When x→a, the functions f(x) and F(x) tend to zero;

(2) In the centripetal neighborhood of point A, both f'(x) and F'(x) exist, and F' (x) ≠ 0;

(3) When x→a, lim f'(x)/F'(x) exists (or is infinite), then

When x→a, lim f(x)/F(x)=lim f'(x)/F'(x).

rebuild

(1) When x→∞, both functions f(x) and F(x) tend to zero;

(2) f'(x) and F'(x) both exist when |x| > is n, and F' (x) ≠ 0;

(3) When x→∞, lim f'(x)/F'(x) exists (or is infinite), then

When x→∞, lim f(x)/F(x)=lim f'(x)/F'(x).

Using L'H?pital's law to find the limit of infinitive is one of the key points of differential calculus. When solving problems, we should pay attention to:

(1) before you start to seek the limit, you should first check whether it meets the 0/0 or ∞ /∞ type, otherwise it is an error of abusing L'H?pital's law. When it does not exist (excluding the ∞ case), it cannot be used. At this time, L'H?pital's law is invalid, and other methods should be used to find the limit. For example, use Taylor formula to solve.

(2) L'H?pital's law can be used repeatedly until the limit is found.

(3) L'H?pital's law is an effective tool to find the limit of indeterminate form, but if only L'H?pital's law is used, the calculation will often be very complicated, so other methods must be combined, such as separating the product factor of non-zero limit in time to simplify the calculation, replacing the product factor with equivalent quantity and so on.