1. Given (a + b) ^ 2 = 7, (a-b) ^ 2 = 4, find the value of a ^ 2 + B ^ 2 and ab 2. Given a (A-1) - A ^ 2-B) = 5, find the value of (B-A) ^ 2 3. It is known that a, B and C are three sides of a triangle, and a ^ 2 + B ^ 2 + C ^ 2-ab-bc-ca = 0

1. Given (a + b) ^ 2 = 7, (a-b) ^ 2 = 4, find the value of a ^ 2 + B ^ 2 and ab 2. Given a (A-1) - A ^ 2-B) = 5, find the value of (B-A) ^ 2 3. It is known that a, B and C are three sides of a triangle, and a ^ 2 + B ^ 2 + C ^ 2-ab-bc-ca = 0

It is known that (a + b) ^ 2 = a ^ 2 + B ^ 2 + 2Ab = 7 (1)
(a-b) ^ 2 = a ^ 2 + B ^ 2-2ab = 4 (2)
(1) Formula + (2) formula A ^ 2 + B ^ 2 = 11 / 2
(1) Equation + (2) gives AB = 3 / 4
2. There are few brackets for the known conditions
3 proof
2（a^2+b^2+c^2-ab-bc-ca）=（a^2-2ab+b^2）+（b^2-2bc+c^2）+（c^2--2ca+c^2）=（a-c)^2+（b-c)^2+(c-a)^2=0
A = b = C, so the triangle is equilateral

1. For a project, team a will finish it in 12 days, team B will finish it in 15 days, and team C will finish it in 20 days. Now team a and team B will work together for several days. In order to speed up the progress, team C will also join in the work. As a result, the project will be completed one day ahead of the original schedule. It is known that the scheduled time is 7 days. How many days did team a and team B cooperate first? How many days did team C do after joining in?
2. The table below records the distribution of the number of people who have scored n goals in a given time. It is known that the average number of people who scored 3 or more goals is 3.5; the average number of people who scored 4 or less goals is 2.5?
The chart is as follows (the question mark is the contaminated place)
The number of goals n: 0 1 2 3 4 5
Number of people who scored n goals: 1, 2, 7, 2

(1) Let Party A and Party B cooperate for X days, and Party C join in for y days
x+y=6
(x+y)/12+(x+y)/15+y/20=1
So x = 4, y = 2
(2) Let's have three X people and four y people
3x+4y+10=3.5x+3.5y+7
2+14+3x+4y=2.5+5+17.5+2.5x+2.5y
So x = 9, y = 3

Given the function f (x) = sin ^ 2x + 2sinxcosx + 3cos ^ 2x, X ∈ [0, π / 2], (1) find the range of the function, and find that the maximum value of the function is the value of X

f(x)=1+sin2x+2cos^2x
=1+sin2x+cos2x+1
=2 + (radical 2) sin (x + Wu / 4)
Because x ∈ [0, π / 2]
So f (x) max = f (Wu / 4) = 2 + radical 2
X = 5 / 4

Y = (cosx-m) ^ 2-1, when cosx = - 1, there is a maximum, when cosx = m, there is a minimum, find the value of M
Given the function y = (cosx-m) ^ 2-1, when cosx = - 1, there is a maximum value, when cosx = m, there is a minimum value, find the value of M
It seems that there are three cases of the axis of symmetry

Let cosx = t, t ∈ [- 1,1], then y = (T-M) & sup2; - 1
The image of the function is a parabola with the opening upward and the symmetry axis t = M
Because there is a minimum value when t = m, so the symmetry axis t = m falls in the domain interval, then there is - 1 ≤ m ≤ 1
If the axis of symmetry falls in the field, then when t ∈ [- 1, M], the function monotonically decreases, and when t ∈ [M, 1], the function monotonically increases, so the maximum value must be at two endpoints, because it is known that when t = cosx = - 1, there is a maximum value
Then y (t = - 1) ≥ y (t = 1)
That is (- 1-m) & sup2; - 1 ≥ (1-m) & sup2; - 1
The solution is m ≥ 0
In conclusion, 0 ≤ m ≤ 1

In order to find the maximum and minimum value in practical problems by using quadratic function, why should we first find out the analytic expression of function and the value range of independent variable
When using quadratic function to find the maximum and minimum value in practical problems, why should we first find out the analytic expression of function and the value range of independent variable? Can we give an example?

For example, some values do not conform to the actual situation, such as money will not appear negative. Such as - 5 yuan and so on
Area will not appear negative value and so on
The reason for finding the domain is that the function must be in the domain to have meaning, otherwise it does not belong to the function
For example, y = root x, X must be greater than or equal to 0 to be meaningful~

The maximum value of function f (x) = 2Sin (3x + π) + 1 is______ ．

∵ f (x) = 2Sin (3x + π) + 1 ∵ sin (3x + π) when 3x + π = 2K π + π 2, that is, x = 2K π 3 − π 6, take the maximum value of 1, then f (x) has the maximum value of 3

How to get the maximum value of sine cosine function by formula
For example, y = cosx + 1, y = - 3sin2x

Sin, cos maximum value is 1, minimum value is - 1, 3sin maximum value is 3, minimum value is - 3, remember this is easy to do

210240270310330, sine and cosine function values of each angle of degree
The sine and cosine functions of these angles

sin210°=sin(180°+30°)=-sin30°=-1/2sin240°=sin(180°+60°)=-sin60°=-√3/2 sin270°=sin(180°+90°)=-sin90°=-1sin300°=sin(360°-60°)=-sin60°=-√3/2sin330°=sin(360°-30°)=-sin30°=-1/2cos210...

Proof: sine function y = SiNx has no positive period smaller than 2 π
It's urgent. It's better to use the counter evidence,

By using the method of proof to the contrary, it is assumed that the sine function y = SiNx has a positive period T, 0 which is smaller than 2 π

The sum of the maximum and minimum of the function y = 1-sinx4 + x2 + 1 (x ∈ R) is______ ．

F (x) = 1-sinx4 + x2 + 1, X ∈ R. let g (x) = - sinx4 + x2 + 1, because g (- x) = - sin (− x) (− x) 4 + (− x) 2 + 1 = sinx4 + x2 + 1 = - G (x), so g (x) is an odd function. The image of an odd function is symmetric about the origin, and its maximum and minimum are opposite numbers. Let m be the maximum value of G (x), then - m be the minimum value of G (x) If the maximum of F (x) is 1 + m, then the minimum of F (x) is 1-m. the sum of the maximum and minimum of F (x) is 2