Find the extreme point of the function obtained by adding the quadratic function and the inverse pro

In this problem, you can use the derivative to determine the monotonicity of the function first, and then derive the minimum value based on the monotonicity.

Let f(x)=x -9 x, and find the derivative of f(x)=2x+9 x =(2x +9) x.

Let f(x)=0, then x=-9 2 under the cubic root number.

Because f(x) is less than zero in the interval (-9 2 under the cubic root number), f(x) is monotonically decreasing in that interval;

Because f(x) is greater than zero in the interval (-9 2,+ under the cubic root number), f(x) is monotonically increasing in that interval;

So when x=-9 2 under the cubic root number, f(x) takes the minimum value, that is, f(x)min=(-9 2) (2 3)+9· (2 9) (1 3)。

Hope it helps!

As mentioned upstairs, this kind of problem is best answered by derivatives. Let the function f(x)=ax +bx, where a and b are constant and not 0, and n is normal. Then the derivative of f(x) is f'(x)=2ax-nbx then let f'(x)=0, i.e., 2ax=nbx x =nb 2a, solve this equation, if n+2 is even, when nb 2a is positive, then there are 2 solutions, and when nb 2a is negative, there is no solution.

If n+2 is odd, then there is 1 solution. On this basis, it is judged whether it is an extreme point.

This type of extreme problem can be solved directly using derivatives. For the above expression, the first derivative of x is obtained, and the derivative is zero, and the coordinates of the extreme point can be obtained.

This student is very important in high school knowledge is functions, so I suggest that you have a good look at the junior high school function part, although this is very preliminary, but it is definitely the foundation part.

Function intersections, I won't tell you how to find them. Let me ask you first, what is a function?

In a Cartesian coordinate system, a function is a graph, a collection of points. At the same time, the function is also a binary equation.

Each pair (x,y) of this binary equation corresponds to a point in the coordinate system.

Two functions have an intersection, which means that there are such (possibly multiple) points that exist on the line corresponding to the two functions at the same time. That is to say (important here, note understanding) that there is such a pair (x,y) of binary linear equations that satisfy both functions at the same time.

It is easy to understand this step, two binary equations to find the same solution, that is, to form them into a system of equations, and solve them in succession.

It needs to be emphasized here that functions and graphs are inseparable, and we should pay more attention to understanding them together, rather than isolating them. When doing problems, although the system of equations can solve the answer, be sure to draw a sketch to deepen your understanding, because sometimes the function solution will miss some special answers.

The inverse function of the proportional function y=kx is y=1 k*x is inversely proportional to the shortform function y=k x is y=k x The inverse function of y=ax+b is y=1 a*x-b a quadratic function y=ax 2+bx+c is y=-b (2a) b 2-4a(c-x)] two) exponential function y=a x is y=log[a].

Teachers should talk about it.

Let's talk about quadratic functions! -2a/b

Refers to the axis of symmetry, which is the abscissa of the vertex.

In addition, it should be b-square-4ac, which is the inverse proportional function in the equation, and what you need to remember is that if it tells you that it is a quadratic function, how do you set it, it is y=k x(k≠0).

In fact, there are so many test points for quadratic functions and inverse proportions, which are nothing more than incrementality, the most worthwhile, intervals!

You have to figure out all these basics, for example, you have to master everything in the vertex style now.

You can go and buy a reference book, which has all of them.

The foundation is always the most critical, and I hope it helps you.

The primary function is a special proportional function The primary function y=kx The proportional function y=kx+b

Quadratic function y=ax +bx+c inverse proportional function y=k x

Inversely proportional hall destruction y=2 x with a one-time function.

y=kx+b is only handed over to the ambush with a point a(1,2), then there is.

2 x=kx+b, i.e., kx 2+bx-2=0 and because it is only handed over to one point, then b 2-4*k*(-2)=b 2+8k=0 and because the function y=kx+b passes the point a(1,2), then the banquet refers to k+b=2 substituting the above equation.

b=4,k=-2, then the primary function is analytical.

is y=-2x+4

Proportional and primary functions: The increase or decrease of the functions y=kx, y=kx+b depends on the constant k. When k 0 y increases with x and decreases with x when k 0y increases. The increment or decrease of a function has nothing to do with b.

2. Inverse proportional function: the inverse column function y=k x is "inverse" compared with the proportional column function and the primary function. When k 0 y decreases with the increase of x, and when k 0y increases with the increase of x.

3. Quadratic function: the increase and decrease of the function is divided by the axis of symmetry, and the vertex is the dividing point.

1) When a 0 is a parabola opening upwards, x -b 2a 0y decreases with the increase of x, and x -b 2ay increases with the increase of x.

2) When a 0 is a parabola opening downward, x -b 2a 0y increases with the increase of x, and x -b 2ay decreases with the increase of x.

f（x）=x^2+4/x

f‘（x）=2x-4/x²

Maximum: f'(x)=0

2x-4/x²=0

2x³-4=0

x = 2 x = 2 cube root.

So when x = the cube root of 2, f(x) may be the cube root with the maximum value at x<2.

f'(x)<0 (subtraction).

x>2.

f’（x）>0

So when x=2 is the cube root, f(x) is the minimum.

You can split the term and use the mean inequality:

f(x)=x +2 x+2 x>=3(x *2 x*2 x) (1 3)=3*4 (1 3), take the equal sign when x = 2 x, that is, take the minimum value when x = 2 (1 3).