Let y root number x 2 4x 13 root number x 2 2x 2 , try to find the minimum value of the function

If you start with y as a distance, the equation may be easier to understand. d = root number (x 2+4x+13) + root number (x 2-2x+2) = root number ((x+2) 2+9) + root number ((x-1) 2+1) = root number ((x+2) 2+(0-3) 2) + root number ((x-1) 2+(0+1) 2). This is the sum of the distances between the point (x, 0) and the point (-2,3) and the point (1,-1).

I don't know if you understand.

We usually calculate the distance from just one point to another, here we find the distance from one point (x,0) to two points. You draw the coordinate axis, the points (-2, 3) and (1, -1) are drawn, the points (x, 0) are on the x-axis, and the straight line between the two points is the shortest, you should know this. The intersection of the points (-2,3) and (1,-1) with the x-axis is the value of x.

x=1/4。

You may ask why the distance is not written as d = root ((x+2) 2+9) + root ((x-1) 2+1) = root ((x+2) 2+(0 + (note that this is a positive sign, the above is a negative sign) 3) 2) + root ((x-1) 2+(0+1) 2), which is the sum of the distances between the point (x, 0) and the point (-2, -3) and the point (1, -1). Both of these points are below the x-axis, so how do you find the distance? So you symmetricize one of the points above the x-axis, and you can see why by drawing a diagram of this.

Because the distance from these two points to (x,0) after symmetry is equal.

y=root(x 2+4x+13)+root(x 2-2x+2)y=root[(x+2) 2+9]+root[(x-1) 2+1]y=root((x+2) 2+(0-3) 2)+root((x-1) 2+(0+1) 2).

That is, the sum of the distances from the point C(X,0) to A(-2,3) and B(1,-1) is required to be the shortest, and only the distance from C to the straight line AB is the shortest.

ab:y+(4/3)x-1/3

d=|0+(4/3)x-1/3|The root number (1 2 + (4 3) 2) = 0, so x = 1 4

y=x 2 under the root number 4x+13+x 2-10x+26 [x-2) 2+9] +x-5) 2+1] [x+7)(x-11)] x-4)(x-6)] Since each term is greater than or equal to 0, in order to get the minimum value of the call, we need to make one of the terms equal to 0, let's look at the different values of x.

x = 7 y = 143

x = 11 y = 42

x = 4 y = 13

x = 6 y = 25

Conclusion, when x = 4 y = x 2 4x+13 + x 2-10x+26 when x = 4 y = x 2-10x+26 is 13

x+y=4 y=4-x (x 2+1)+ y 2+4)= x 2+1 2 [(x-4) 2+2 2] The above equation can be seen as the sum of the distances from point a(x,0) to point b (lead gear 0,1) and point c(4,2). On the coordinate axis, the minimum value of the above equation is to find the distance from one point to two points on the x-axis and the minimum of the cherry blossom to find the pair of c on the x-axis.

Let the root number (x 2+3) = t, t is greater than the grinding god and equal to the root number 3

y=(t 2+1) t=t+1 t increases in the blind luck deficit domain of the definition of t, so y is greater than equal and quietly shuddering at 4 root number 3=4 * root number 3 3

Summary. y= [x-2) +0+1) ] x-1)+(0-3)] y is the distance from p(x,0) to a(2,-1) and b(1,3) and so apb collinear minimum min=}ab|=√1²+4²)=17

Find the function y = the minimum value under the root number (x 2-4x+5) + under the root number (x 2-2x+10).

y= [x-2) +0+1) ] x-1)+(0-3)] y is the distance from p(x,0) to a(2,-1) and b(1,3) and so apb collinear minimum min=}ab|=√1²+4²)=17

No. Take a photo of the original question and send it over.

y= (x 2+4) + x-1) 2+1] can be seen as the sum of the distances from p(x,0) to a(0,2) and b(1,1) on the x-axis. Remembering the symmetry point a'(0,-2) of a with respect to the x-axis, then pa+pb=pa'+PB is based on the principle of the shortest line segment between two points, when P is A'When the balance is blind and the intersection of the B and the x-axis, the minimum value is bent. And a'b= empty(1 2+(-2-1) 2) = 10 i.e.

I forgot the method of high school for a long time, so I will introduce you to the method of high mathematics.

Conditional equation: x+y-12=0

Find the maximum-value equation: (x 2+4) + y 2+9) then the Lagrangian equation is l(x,y) = x 2+4) + y 2+9) + k(x+y-12) [k is a real number].

l(x,y) for x is the partial derivative x (x 2+4)+kl(x,y) for y (y 2+9)+k (x 2+4)+ y 2+9) The maximum and minimum values are both good pose 0

That is, x (x 2+4) + k = y (y 2 + 9) + k = 0 then x (x 2+4) = y (y 2 + 9) is brought into the conditional equation y=12-x

The solution is x1=24 5, x2=-24

Next, x1 and x2 are sent back (x 2+4)+ y 2+9) respectively It can be seen that when x=24 5 is the minimum value of the dust sock, then the minimum value is 13

Solution: 1y= (x-0) 2+(0-1) 2) +x-4) 2+(0-2) 2).

Represents the sum of the distances from points to (0,1) and (4,2) on the x-axis.

According to the principle of optical path, when the point (0,1) and the point (4,2) have the same reflection angle with the x-axis, the light travels the shortest distance, that is, the smallest y.

1/x=2/(4-x)

x=4 3Substituting x=4 3 into the equation.

y small = 5 3 + 10 3 = 5

Solution: 2y= (x-0) 2+(0-1) 2) +x-4) 2+(0+2) 2).

Represents the sum of the distances from points on the x-axis to (0,1) and (4,-2), and obviously, the two points are connected the shortest.

y small = (0-4) 2 + (1 + 2) 2) = 5 solution to talk about brother such as 1

y=√(x-0)^2+(0-1)^2) +x-4)^2+(0-2)^2)

Represents the sum of the distances from points to (0,1) and (4,2) on the x-axis.

According to the principle of optical pathlength, when the reflection angle of the point (0,1) and the point (dust scale 4,2) is the same as that of the x-axis, the distance traveled by the light is the shortest, i.e., the smallest y.

1/x=2/(4-x)

x=4 3Substituting x=4 3 into the equation.

y small = 5 3 + 10 3 = 5

Solution: 2y= (x-0) 2+(0-1) 2) +x-4) 2+(0+2) 2).

Represents the sum of the distances from points on the x-axis to (0,1) and (4,-2), and obviously, the two points are connected the shortest.

y small = (0-4) 2+(1+2) 2)=5 satisfaction, remember to add more points!

[Ordered the desired formula m].

m=√(x^2+4)+√y^2+9)

[(x-0)^2+(0-2)^2] +x-12)^2+(0-3)^2]

Distance from point (x,0) to point (0,2) + distance from point (x,0) to point (12,3).

That is, the distance from a point to a point (0,2) and a point (12,3) on the x-axis is taken and (0,2) is taken with respect to the symmetry point (0,-2) on the x-axis

Connecting (0,-2) (12,3), the distance between the two points is the minimum value of the equation 13, where x=24 5, y=36, 5