Difficulty 1: the function runs through the whole senior three. Senior one learns basic elementary functions, senior two learns functions and derivatives, and the function ideas and methods can be used in many other knowledge points. Function accounts for about 30% of the scores in the college entrance examination mathematics, and its importance can be imagined. The difficulty lies in understanding, many people can't grasp its abstraction and change, and as a derivative of the finale, few people can do it.

Solution: Indeed, function is a main thread that runs through the whole middle school mathematics, and its content includes two aspects: nature and image. The expansion of function knowledge mainly combines equations (zeros) and inequalities. The dominant idea of dealing with these two kinds of problems is transformation, and the direction of transformation is solved by means of the properties of images and functions. On the path of transformation, we have developed the ∞ graph of function problem-solving thinking, which can be said for sure that the thinking path of all problems of function.

Difficulty 2: derivative. Derivative, as an important test content of NMET mathematics, often appears as a finale in NMET, and the difficulty of the test questions increases year by year. It is difficult to prove functional inequality as derivative, which makes many candidates flinch. Among them, there are three kinds of functional inequality problems in NMET finale in recent years, one is to hide zero, the other is to deviate from double zero or extreme point, and the other is to assign the existence of zero.

Solution to the hidden zero problem: It is proved that functional inequalities are often transformed into monotonicity or maximum value of functions, which involves monotonicity, extreme value and maximum value, which in turn involves the zero point problem of derivative functions. If the zero point of the derivative function cannot be found, we call it the hidden pole problem or the hidden zero point problem. In this respect, the investigation of the national grand finale is often innovative through continuous inheritance.

For the hidden zero problem, the structural characteristics of the topic often show the mixed form of exponential function, logarithmic function, trigonometric function and power function. The reason why the hidden zero is introduced is that the derivative zero cannot be found. After introducing the hidden zero, the following conversion principle can be summarized as "pointing to the triangle power" in seven words, that is to say, exponential structure, logarithmic structure and triangular structure are all converted into power functions. Investigate its root cause. Because power function is our good friend and our most familiar partner (its higher background is Taylor formula). After conversion, it is often necessary to evaluate with the zero theorem, and finally deal with it as a whole.