● Difficult magnetic field
(★★★★★★) Let f (x) = log2 and f (x) =+f (x).
(1) Try to judge the monotonicity of function f(x), define it by monotonicity of function, and give proof;
(2) If the inverse function of f(x) is f- 1(x), it is proved that for any natural number n(n≥3), there exists f-1(n) >:
(3) If the inverse function F- 1(x) of F(x), it is proved that the equation F- 1(x)=0 has a unique solution.
● Case study
[Example 1] It is known that the straight line passing through the origin o intersects with the image of function y=log8x at points A and B, and the parallel line passing through points A and B with the Y axis intersects with the image of function y=log2x at points C and D. 。
(1) Prove that points C, D and origin O are on the same straight line;
(2) When BC is parallel to the X axis, find the coordinates of point A. 。
Proposition intention: This topic mainly examines the basic knowledge such as logarithmic function image, logarithmic formula, logarithmic equation and exponential equation. To examine students' analytical and operational abilities. Belong to the category of ★★★★.
Knowledge support: (1) Proof method of three-point * * * line: kOC=kOD.
(2) The solution of the problem (2) contains the idea of the equation. As long as the equation (1) is obtained, the coordinates of point A can be obtained.
Wrong solution analysis: it is not easy to consider solving practical problems with equation thinking.
Skills and methods: the first question of this question proves the three-point * * * line with equal slope; The second question is to find the coordinates of point A by using the idea of equation.
(1) It is proved that the abscissas of point A and point B are x 1 and x2 respectively. 1, x2> 1, then the ordinate of a and b are log8x 1 and log8x2, respectively. Because A and B are on a straight line passing through point O, the coordinates of point C and point D are (x 1, log2x 1) and (x2, log2x2) respectively. Because log2x 1,
The slope of OD: k2=, so it can be seen that k 1=k2, that is, O, C and D are on the same straight line.
(2) Solution: From BC parallel to the X axis, we know: log2x1= log8x, that is, log2x 1= log2x2, and substitute it into x2log8x1= x13log8x1= 3x65438. 1 Know log8x 1 ≠ 0, ∴ x 13 = 3x 1. And x1>; 1,∴x 1=, then the coordinate of point a is (,log8).
There are a series of points P 1 (A 1, B 1), P2 (A2, B2), ..., Pn (An, bn) ... for each natural number, the n-point Pn is located in the function y=2000( )x(0.
(1) Find the expression of the ordinate bn of the point Pn;
(2) If a triangle can be formed for each natural number n, BN, BN+ 1 and BN+2, find the range of a;
(3) Let Cn=lg(bn)(n∈N*). If a takes the smallest integer in the range determined in (2), what is the maximum sum of the first term in the sequence {Cn}? Try to explain why.
Proposition intention: this question combines the knowledge points of plane point series, exponential function, logarithm, maximum value and so on, forming a comprehensive question with great difficulty in thinking. This question mainly examines the examinee's ability to analyze and apply comprehensive knowledge, which belongs to the level of ★★★★★★
Title.
Knowledge support: knowledge of exponential function, logarithmic function, sequence and maximum value.
Misunderstanding analysis: it is not easy for candidates to master comprehensive knowledge, have difficulty in thinking and can't find a breakthrough in solving problems.
Skills and methods: This question belongs to comprehensive knowledge, and the key lies in thinking and understanding the conditions in the process of examining the question and using relevant knowledge points to solve the problem.
Solution: (1) From the meaning of the question: an=n+ ,∴bn=2000 ().
(2)∵ function y = 2000 () x (0bn+1>; Bn+2。 If BN, BN+ 1 and BN+2 are side lengths, a triangle can be formed if and only if bn+2+bn+1> Bn, that is, () 2+()-1>; 0, the solution is a5(-1). ∴5( - 1).
(3)∵5( - 1)
∴bn=2000()。 The sequence {bn} is a decreasing positive sequence, and for each natural number n≥2, Bn=bnBn- 1. So when bn≥ 1, bn
● Roll up your sleeves
The problems and solutions involved in this difficulty are:
(1) Use the images and properties of two functions to solve basic problems. This kind of questions requires candidates to master the images and properties of functions and use them flexibly.
(2) Comprehensive topics. This kind of topic requires candidates to have strong analytical and logical thinking skills.
(3) Application topics. This kind of topic requires candidates to have strong modeling ability.
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