Question 1: This kind of topic is called monotonicity of composite function. 2X-X square is considered as G(X)=2X-X square. Monotonicity of unary function means whether f(x) increases or decreases when x increases. So first find out the monotone interval of G(X) on the domain (the domain must be found, and the domain of this problem is R). For example, this problem G(X) is in (-infinity, 1), and G(X) is due to the known monotonic decrease of F(X). When X is in (-infinity, 1), the increase of X leads to the decrease of G(X), while the decrease of G(X) leads to the decrease of F (. That is to say, G(x) acts as a bridge process, that is, when X is at (-infinity, 1), the increase of X will eventually lead to the increase of F(x), that is, it will increase monotonically. Please analyze the monotone decreasing interval by yourself.

The way to solve this kind of problem is basically to see through the bridge function of the composite function G(x). The essential problem is to see how to change F(x) through some bridges when x increases. Because it is convenient for the landlord to understand, it is specially explained in popular language. I hope that the landlord will draw inferences from others. Experience the methods and ideas in mathematics by yourself.

The second question: first, we should use common language to clarify the idea of the landlord. I wonder if the landlord can think of a function he learned in junior high school when he saw this formula:

F(x)=-2/3X。 This function meets all the requirements in the second question, which can be said to be a special case in the second question, but first of all, we must never think that F(x) is -2/3X, and we must never confuse the relationship between general and special. This example is given here to compare the abstract mapping relationship with the image function relationship, which is convenient for thinking. Obviously, it is a monotonically decreasing function. If the topic goes deeper in the future, ask the landlord for monotonicity first, and then let the landlord prove it. I hope the landlord can guess with the model. Then use the following method to prove it.

Because of monotonicity: let X 1X2 belong to R, then X 1=X2+a(a0).

f(x 1)-f(X2)= f(x2+a)-f(x2)= f(a)0

So monotonously decreasing.

To sum up the judgment of monotonicity for the landlord, we can see that the change of X will eventually lead to the change of F(X). Evidence about him. Let's use subtraction. Please forgive me for talking too much and help the landlord clear his mind. If you have any questions, welcome to discuss. QQ7 19 144797