Given that a and B are real numbers, the function f (x) = x ^ 2 + BX + C has the following properties for any α, β∈ R: F (sin α) ≥ 0, f (2 + cos β) ≤ 0 (1) Finding the value of F (1) (2) It is proved that C ≥ 3 (3) Let the maximum value of F (sin α) be 10, and find f (x) Request detailed process

Given that a and B are real numbers, the function f (x) = x ^ 2 + BX + C has the following properties for any α, β∈ R: F (sin α) ≥ 0, f (2 + cos β) ≤ 0 (1) Finding the value of F (1) (2) It is proved that C ≥ 3 (3) Let the maximum value of F (sin α) be 10, and find f (x) Request detailed process

f(x)=x^2+bx+c
For any α, β ∈ R, we have:
f(sinα)≥0
f(2+cosβ)≤0
α=90,f(1)>=0
β=180,f(1)

M is a real number. The square of equation 5x minus 12x plus 4 plus m equals 0. If one root is greater than 2 and the other root is less than 2, the value of M is obtained

5x^2-12x+4+m=0
There are two roots, so 144-4 * 5 * (4 + m) > 0 so m2
12 radical [144-4 * 5 * (4 + m)] / (10) 64M] > - 8
Forget it, I can't do it if it's right

It is known that there are two equations X & # 178; + ax + A & # 178; - 1 = 0 about X, and both of them are greater than - 1, then the value range of real number a is obtained

If the equation has real roots, then Δ = a ^ 2-4 (a ^ 2-1) ≥ 0 = = > A ^ 2 ≤ 1 / 3 = = > - √ 3 / 3 ≤ a ≤√ 3 / 3
Let X1 and X2 be the two parts of the equation respectively, then
x1＞－1,x2＞－1
∴x1＋1＞0,x2＋1＞0
Then (x1 + 1) + (x2 + 1) > 0 = = > X1 + x2 + 2 > 0
(x1＋1)*(x2＋1)＞0 ===>x1x2＋(x1＋x2)＋1＞0
From the relationship between root and coefficient, we can know that:
x1＋x2＝－a,x1x2＝a^2－1
The above formula can be changed into
－a＋2＞0 ====>a＜2
A ^ 2-1-a + 1 > 0 = = > a > 1 or a < 0
A < 0 or 1 < a < 2
The range of a is (- ∞, 0) ∪ (1,2)

It is known that circle C: X & # 178; Y & # 178; - 2ax-4y A & # 178; = 0 (a > 0) and line L: X-Y 3 = 0 when line L is cut by circle C
When the chord length is 2 root sign 2

The center C coordinate is (a, 2) and the radius is 2,
The distance between C and line l d = |a-2 + 3 | / √ 2
According to (chord length / 2) ² = R & #178; - D & #178;, we can get
3=4-[(a+1)²/2]
That is (a + 1) ² = 2
That is, a = - 1 ± √ 2