Special topic II on mathematical physics of the sprint strong foundation program

1. Strong base program mathematical and physical simulation volume

I have worked out a set of mathematical and physical simulation test questions for the strong foundation plan , It is suitable for college entrance examination students with a certain foundation . Due to my limited level , If there are omissions, please criticize and correct them ！

2. Inequality topics

2.1 Qin Sheng inequality （ A combination of numbers and shapes ）

2.1.1 The first question of the third question in the simulation volume

Known functions Satisfy . Might as well set . among $a < p> <>

It is easy to get from the graph relation , trapezoid 、 trapezoid 、 Curved edge trapezoid The area of ( Curved trapezoid by ) Satisfy ：

So there is ：

It can be obtained. ：

2.1.2 The second question of the third question in the simulation volume

Want to use Qin Sheng inequality , The left side is easily reduced to the form of function mean , But the right is not a function of the arithmetic mean , It's a function of the geometric mean . therefore , Consider logarithmic transformation , Convert geometric mean to arithmetic mean .

Make . Make . It's easy to know , When when ,, And only if when ,. therefore , From Qin Sheng's inequality ：

That is to say ：

The original inequality can be obtained by further simplification .

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2.2 Trigonometric inequality

2.2.1 The first question of the fourth question in the simulation volume

Make , Then there are ：

Make , By You know ,

Make , be . So there is . So there is

therefore ,

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2.2.2 The second question of the fourth question in the simulation volume

Acute triangle in , Yes ：

namely ：

Simplify to ：

It's easy to know , For the function , among , Yes . Therefore, from Qin Sheng's inequality ：

So there is ：

If and only if when , The equal sign is established .

2.2.3 The third question of the fourth question in the simulation volume

stay There are . From Qin Sheng's inequality ：

namely ：

From the mean value inequality ：

So there is ：

If and only if when , The equal sign is established .

2.3 mean value inequality & Cauchy inequality

2.3.1 Corresponding to the first question of the fifth question in the simulation volume

Investigate the matching technique of mean inequality

2.3.2 Corresponding to the second question of the fifth question in the simulation volume

Investigate the matching technique of mean inequality

2.3.3 come from 2021 Peking University strong foundation program mathematics test questions

subject ： If the real number Satisfy , be The minimum value of is ？

answer ： It is known that , Then there are ：

So the idea is to Put together , Put together .

The first inequality uses the fractional Cauchy inequality , If and only if And The equal sign is established . The second inequality uses the mean inequality , If and only if The equal sign is established .

Sum up , The minimum value is 2, And only if or Get the minimum .

2.3.4 come from 2021 Shanghai Jiaotong University strong foundation plan mathematics test paper

subject ： It is known that Is a positive number , seek The minimum value of .

answer ： Observe to get , To obtain the minimum value of Must be satisfied with ：

therefore , The method of undetermined coefficient is used for mean inequality fitting ：

If you want to use the mean inequality , We must have , In order to make sure The previous coefficients are the same . Besides , And make sure that Is proportional to the coefficient of , And the proportion is . So there is ：

Solution ,, be .

therefore , You can answer as follows ：

If and only if The equal sign is established .

therefore ,

If and only if Get the minimum .