1.X+Y/2+Z/3=(X+Y/2+Z/3)*1=(X+Y/2+Z/3)*(1/X+2/Y+3/Z)=1+2X/Y+3X/Z+Y/2X+1+3Y/2Z+Z/3X+2Z/3Y+1=3+(2X/Y+Y/2X)+(3X/Z+Z/3X)+(3Y/2Z+2Z/3Y)
因為X,Y,Z∈R+,所以可用均值不等式
3+(2X/Y+Y/2X)+(3X/Z+Z/3X)+(3Y/2Z+2Z/3Y)>=3+2+2+2=9
所以X+Y/2+Z/3的最小值是9
等號成立的條件是2X=Y,3X=Z,3Y=2Z,即X=3,Y=6,Z=9時等號成立
所以最小值就是9
2.利用不等式2/[(n)^(1/2)+(n+1)^(1/2)]<n^(1/2)<2/[(n-1)^(1/2)+(n)^(1/2)],即
2[(n+1)^(1/2)-n^(1/2)]<n^(1/2)<2[(n)^(1/2)-(n-1)^(1/2)]
進行裂項相消可得:
1+2[100^(1/2)-2^(1/2)]=1/100^(1/2)<s<1+2[100^(1/2)-1^(1/2)]
即:18.182<s<19
故,s的整數部分是18
3.0<x<1/2
x>0,1-2x>0
所以[x*x*(1-2x)]的立方根<=[x+x+(1-2x)]/3
[x?(1-2x)]的立方根<=1/3
x?(1-2x)<=1/27
4.M=(a+b+c)(a?+b?+c?)≥×3√abc×3√a?b?c?=9abc=N
(這裏√表示開3次方)
5.1=x^2+2y^2=x^2+y^2+y^2>=3*(x^2*y^2*y*2)^(1/3)=3(x^2*y^4)^(1/3)
(x^2*y^4)^(1/3)<=1/3
x^2*y^4-1<=(1/3)^3=1/27-1=-26/27
所以最大值是
=-26/27