In recent years, scaling method often appears in college entrance examination questions, but most students have no way to start. The essence of scaling method: to prove the inequality A < B, sometimes one side can be enlarged or reduced to find an intermediate quantity, such as enlarging A to C, that is, A < C, and then proving C < B. ..
The common skills of scaling are: (1) Leave out (or add) some items. (2) Enlarge or reduce the numerator or denominator in the fraction. (3) Scaling by applying basic inequalities. (4) Monotonicity of the function is applied to scaling. (5) Scale according to the topic conditions. The following is an example by the author.
First, delete (or add) some items.
Example: In the known series, prove:
Proof: when k = 2,3 ... there is, so, Ⅷ.
It's ∴∴∴ again, so it's certified.
Note: Discard (or add) some items, that is, add some positive values to the polynomial, and the value of the polynomial will become larger, and subtract some negative values from the polynomial, and the value of the polynomial will become smaller.
Second, enlarge or reduce the numerator or denominator in the fraction.
Example: Verify:
Analysis: The desired formula is in the form of sum, but it cannot be solved by sum formula. The denominator can be appropriately enlarged or reduced to a summation formula, and then summed.
Proof: ∵ When appropriate,
∴, that is, let K=2, 3 respectively. . . . Get. . . ,
Add these equations: namely:
Therefore, it has been certified.
Thirdly, the application of basic inequality scale.
Example: Set up and verify:
Prove: obviously and
Therefore-①
- ②
Therefore, it has been certified.
Note: ① Scale with basic inequality, ② Scale with some terms.
Fourthly, the monotonicity of scale application function.
Example: It is known that this proof holds for any inequality.
Prove that it is obviously a decreasing function in the world, and there is a maximum value M= and a minimum value N= in the world.
Ⅶ of arbitrary constant
Fifth, scale according to the theme conditions.
For example, a quadratic function is known, in which sum
(1) Verification: The image of this function intersects the X axis at two different points.
(2) Let the length of the function image passing through the X axis be, and verify that:
Analysis: The problem (1) belongs to the basic problem and can be transformed by the actors. Question (2) requires the establishment of a functional relationship between and.
(1) Proof: ∫∴
∴△
∴ The image of this function intersects the X axis at two different points.
(2) Let the two intersections of the image of the function and the X axis be respectively, then
You know:
∴, and it is a monotone decreasing function in the world. ∴ 。