Cauchy inequality mentioned in this paper refers to
(I = 1, 2, ..., n) (1) The equal sign holds if and only if. This is also holder's inequality (where k> 1, k/> 1, and,,, I= 1, 2, ..., n). When k=2, k/=2.
There are many ways to prove inequality (1), and the methods acceptable to middle school students are collocation, discriminant, mathematical induction, etc., so I don't need to repeat them here. Let's talk about its application in middle school mathematics.
Derive important formulas
1, proving that the square average of n real numbers is not less than the arithmetic average of n numbers, that is, if, then (2)
Proof: by Cauchy inequality
therefore
Therefore, when n=2 in formula (2), this is the 1 1 question in the middle school mathematics textbook (Volume II).
Inequality (2) extends the only problems of "arithmetic average" and "geometric average" in middle school textbooks to "quadratic power average", which not only broadens the horizons of middle school students, but also opens up new ways to solve many inequality problems.
2. Derive the distance formula from point to straight line, that is, the distance from point P (x0, y0) to straight line L: ax+by+c = 0.
The above non-strict inequality only takes the equal sign when B(x 1-x0)=A(y 1-y0), that is, pq ⊥ l.
Therefore, the formula is proved.
Prove inequality
It is especially convenient to prove some inequalities with Cauchy inequality. There are many such questions in the current high school textbooks, such as 1 1, the fifth review question of P32 in the second volume of high school algebra: it is known and verified, and the beginning of the work proved by Cauchy inequality is clear at a glance, so it is very convenient to prove it by Cauchy inequality. Another example is P 16, the first question 19: A, B, c∈R+ is known, and it is simply proved as Cauchy inequality, left =. Get a license.
Then prove an inequality with trigonometric function.
Let a, b, c>0 and acos2θ+bsin2θ < C, and verify.
Proof: From Cauchy inequality and problems, we can get
therefore
Find the maximum value
Using Cauchy inequality, we can easily solve the maximum or minimum problems of some functions.
Example 2 the maximum values of a, b, c∈R+ and a+b+c= 1 are known.
Solution: obtained from Cauchy inequality
If and only if the equal sign holds.
Therefore, the maximum value of is.
The minimum value in Example 3.
Solution: obtained from Cauchy inequality
Therefore, if and only if the equal sign holds. therefore
Four applications in geometry
The heights of the three sides A, B and C of a triangle are ha, hb, hc and R, and the radius of the inscribed circle of this triangle. If R, try to judge the shape of this triangle.
Solution: If the area of the triangle is S, then 2S=aha=bhb=chc, because 2S=r(a+b+c), ha, +hb+hc.
From Cauchy inequality:
Take the equal sign if and only if a=b=c b = c b = c.
Therefore, ha, +hb+hc≥9r is an equal sign if and only if A = B = CB = C. So, when ha, +hb+hc=9r, the triangle is an equilateral triangle.
△ABC's three sides are A, B and C, and its circumscribed circle radius is R, which proves that:
Proof: It is obtained from sine theorem in triangle, so, similarly,
So left =
So the original inequality was proved.
As can be seen from the above examples, Cauchy inequality is not only a very important inequality in higher mathematics, but also plays a great guiding role in elementary mathematics. Using it, we can be far-sighted and condescending, thus solving some related problems in middle school mathematics conveniently.
The following questions are for readers to practice:
Given x, y, z∈R+ and x+y+z= 1, prove it.
Given 3x2+2y2≤6, find the maximum value of 2x+y (A).
Given a+b+c+d+e=8, a2+b2+c2+d2+e2= 16, find the maximum value of E .. (1)
Seek proof
Let p be a point in △ABC, a 1, a2 and a3 are the lengths of three sides of a triangle, r 1, r2 and r3 represent the distances from P to three sides respectively. It is proved that when p is the heart of △ABC, the minimum value is taken.