2. In the triangle ABC, the known vector AB* vector AC= 1, vector AB* vector BC=-2( 1) find the length of AB; (2) Proof: tanA = 2tanB(3) If vector AC = 2, find vector BC (│ is an absolute sign).
3. What is the minimum positive period of the function f (x) = (cosx) 4-(sinx) 4+ 1?
4. What is the maximum value of the function y=5* in root number (x- 1)+ root number (10-2x)?
The process of solving problems.
1.( 1) A4=A 1+3d=-3，a 1 =-3( 1+d)……①，(a 1-2)(a 1+4d)=(a 1+2d)？ ...②，d = 1(∵d >； 0)
(2)∵a 1 =-3-3d =-6 & lt； 0，d = 1 & gt； 0, ∴ Sn has a minimum value, that is, An=n-7≤0,
A(n+ 1)=n-6≥0, 6≤n≤7, ∴ n=6, that is, the sum of the first 6 or 7 items of the sequence {an} is the smallest, and the minimum value is S6 = S7 = (-6) × 6+6× 5/2 =-2655.
2.( 1) Let a (0 0,0, b (a a,0) and c (c, b), then vector AB = (a,0), vector AC=(c, b), vector BC = (c-a, b). ∫(a)= 3，∴ |AB|=|a|=√3
(2) cosA= 1/(|AB||AC|)，cos(π-B)=-2/(|AB||BC|)，0.5 | AB | | AC | sinA = 0.5 | AB | | BC | sinB = = = & gt； tanA=2tanB
(3) cosA= 1/(2√3), and |BC| is obtained from cosine theorem? =5，∴ |BC|=√5
3.f(x)=(cosx)^4-(sinx)^4+ 1=(cos？ X+ crime? x)(cos？ x-sin？ x)+ 1
= 1+cos2x, ∴ Minimum positive period T=π
4.∵ Definition domain x∈. ∴ - 1≤(x-3)/2≤ 1，
y？ = 23x- 15+20√2 √[ 1-(x-3)？ /4], let x-3=2sinθ, then x=3+2sinθ.
-π/2≤θ≤π/2，y？ = 23 (3+2 sin θ)-15+20 √ 2 cos θ = 46 sin θ+20 √ 2 cos θ+54 = 54 sin (θ+φ)+54 (where tan φ = 10 √ 2/23), ∫
∴ y？ ≤ 108, y≤6√3, and the maximum value is 6√3.