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Senior high school refined mathematics answer
1.y = X+ 1/X & gt; =2√(x* 1/x)=2, and the minimum value =2.

2. Let the side length x, the side L=2X, and the included angle a of the regular triangle at the bottom.

Height of regular triangle H=Xsin60 degrees =√3/2X.

Height of the outside of isosceles triangle h=√(L? -(x/2)? )=√[(2x)? -(x/2)? ]=√(4x? -x? /4)=√ 15x/2

cosa=(H? +L? -h? )/(2HL)=(3/4x? +4x? - 15/4x? )/(2√3/2x * 2x)= 1/(2√3)=√3/6

3.f(x)= a b = 1 *( 1+sin2x+cos2x)+ 1 *( 1)= sin2x+cos2x = sin2x+sin(π/2-2x)

= 2 sinπ/4 cos(2x-π/4)=√2 cos(2x-π/4)

Domain: r, range [-√2, √2], minimum positive period =2π/2=π.

4.3Sn=5an-an- 1+3Sn- 1,3(Sn-Sn- 1)= 5an-an- 1,3an=5an-an- 1

2an = an- 1, an/an- 1 = 1/2 equal ratio series.

( 1)an=a 1q^(n- 1)=2*( 1/2)^(n- 1)=2^(2-n)

(2)b 1=2^ 1=2 bn=4/an=2^n

tn=b 1(q^n- 1)/(q- 1)=2(2^n- 1)/(2- 1)=2(2^n- 1)

5.2a=√2,a=√2/2

e=c/a=c/(√2/2)=√2/2,c= 1/2

Answer? -B? =c? , 1/2-b? = 1/4,b? = 1/4,b= 1/2

( 1) x? /( 1/2)+y? /( 1/4)= 1,2x? +4y? = 1

(2) Right focus (c, 0), namely (1/2, 0)

L:y=x+b,0= 1/2+b,b=- 1/2

y=x- 1/2

Substitute into elliptic equation: 2x? +4(x- 1/2)? = 1,2x? +(2x- 1)? = 1

2x? +4x? -4x+ 1= 1,6x? -4x=0,x 1=0,x2=2/3

y 1=- 1/2,y 1 = 2/3- 1/2 = 1/6

A(0,- 1/2),B(2/3, 1/6)

AB=√[(0-2/3)? +(- 1/2- 1/6)? ]=√[4/9+4/9]=2√2/3

6. Assuming an increase of X yuan, the sales volume will decrease by 10X, and the actual sales volume will be 500- 10X.

And the profit of each piece =(50-40)+X= 10+X yuan.

Total profit y = (10+x) (500-10x) =10 (10+x) (50-x) =10 (500+40x-x). )= 10(500+400-400+40X-X? )

= 10(900-(20-X)? )

When X=20, y max = 10*900=9000 yuan.