This is the Yueyang City junior high school mathematics competition question I saw on today’s headlines. Competition questions may be difficult questions in everyone's mind, but here you will find that the competition questions are basic questions. The question of
is as follows: If the real numbers x and y satisfy 3x+4y=-15, find the maximum value of xy.
Yueyang Junior High School Mathematics Competition question
∵3x+4y=-15, ∴x=-4y/3-5, substitute xy:
xy=(-4y/3-5)y=-4y²/3-5y,
this It is a quadratic function , opening downward and achieving the maximum value at the vertex. We can just find the vertex coordinates.
When y=-b/2a=-15/8, the maximum value of
xy=-4(-15/8)²/3-5(-15/8)=75/16.
was built very quickly. It’s also easy to understand.
I have always emphasized that it is very important to learn the basic concepts of mathematics well. After laying a good foundation in mathematics, you can look at mathematical models and problem solving routines to prepare for the exam.
Mathematical models and problem solving routines are not necessary, but learning the basic concepts of mathematics is necessary.
Now let’s look at the teacher’s answer. Teacher
uses basic mathematical inequalities in mathematics to solve the problem. This is also a basic mathematical concept that needs to be mastered. I hope everyone can use it skillfully.
Teacher’s answer
Let’s take a look at the use of basic inequalities.
∵(a-b)²≥0,
∴a²+b²≥2ab, or ab≤(a²+b²)/2.
only takes the equal sign when a=b. This is the basic inequality. When
actually does the questions, the basic inequalities have some deformations and need to be used flexibly. For example, this question is like this.
∵a²+b²≥2ab,
∴a²+2ab+b²≥4ab,
(a+b)²≥4ab, or ab≤(a+b)²/4. How to use this basic inequality in the competition question
?
3x is regarded as a, and 4y is regarded as y, that is, a=3x, b=4y.
3x×4y≤(3x+4y)²/4=225/4,
xy≤75/16, so the maximum value of xy=75/16.
is here to learn mathematics in an easy and simple way, master the basic concepts, and the difficult problems are not a problem.
