a,b属于正实数,a+b=1,求证1/a+1/b+1/ab>=8还有（a+1/a)^2+(b+1/b)^2>=25/2

a,b属于正实数,a+b=1,求证1/a+1/b+1/ab>=8还有（a+1/a)^2+(b+1/b)^2>=25/2
a,b属于正实数,a+b=1,求证1/a+1/b+1/ab>=8
还有（a+1/a)^2+(b+1/b)^2>=25/2

a,b属于正实数,a+b=1,求证1/a+1/b+1/ab>=8还有（a+1/a)^2+(b+1/b)^2>=25/2
1/a+1/b+1/ab
=1/a+1/b+(a+b)/ab
=1/a+1/b+(1/b+1/a)
=2(1/a+1/b)
=2[(a+b)/a+(a+b)/b)]
=2[1+b/a+a/b+1]
=2[2+(b/a+a/b)]
≥2[2+2]
=2*4
=8
1/a+1/b+1/ab>=8
a^2+b^2≥(a+b)^2/2=1/2
ab≤[(a+b)/2]^2=1/4
1/ab≥4
1/a^2+1/b^2≥2/ab≥2*4=8
（a+1/a)^2+(b+1/b)^2
=a^2+1/a^2+2+b^2+1/b^2+2
=4+(a^2+b^2)+(1/a^2+1/b^2)
≥4+1/2+8
=25/2

设a=sin(x)^2
b=cos(x)^2
1/a+1/b+1/(ab)=2/(sin^2*cos^2);sincos<=0.5;1/a+1/b+1/ab>=8
a+1/a)^2+(b+1/b)^2=4+sin^4+cos^4+1/sin^4+1/cos^4>=4+2sin^2+cos^2+2/(sin^2*cos^2)
由于sin^2cos^2<=0.25

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设a=sin(x)^2
b=cos(x)^2
1/a+1/b+1/(ab)=2/(sin^2*cos^2);sincos<=0.5;1/a+1/b+1/ab>=8
a+1/a)^2+(b+1/b)^2=4+sin^4+cos^4+1/sin^4+1/cos^4>=4+2sin^2+cos^2+2/(sin^2*cos^2)
由于sin^2cos^2<=0.25
所以当sin^2cos^2=0.25时4+2sin^2+cos^2+2/(sin^2*cos^2)取最小，即sin^2=cos^2=0.5此时4+2sin^2+cos^2+2/(sin^2*cos^2)=25/2
注：
x+1/x的单调性：
f(x)=x+1/x
f'=1-1/x^2当 -1所以sin^2cos^2+1/(sin^2*cos^2)当sin^2*cos^2取最大值0.25时最大

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1, 1/a+1/b=(a+b)/ab=1/ab
1/a+1/b+1/ab=2/ab
1=a+b>=2根号（ab) 根号（ab) <=1/2 ab<=1/4
2/ab >=8 所以 1/a+1/b+1/ab>=8
2，（a+1/a)^2+(b+1/b)^2＝a*a+2+1/a*a + b*b+2+1/b*b<...

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1, 1/a+1/b=(a+b)/ab=1/ab
1/a+1/b+1/ab=2/ab
1=a+b>=2根号（ab) 根号（ab) <=1/2 ab<=1/4
2/ab >=8 所以 1/a+1/b+1/ab>=8
2，（a+1/a)^2+(b+1/b)^2＝a*a+2+1/a*a + b*b+2+1/b*b
=a*a+b*b+1/a*a+1/b*b+4
=(a+b)(a+b)-2ab+(a*a+b*b)/(a*a*b*b)+4
=1*1-2ab+4+(a*a+b*b)/(a*a*b*b)
=5-2ab+[(a-b)(a-b)+2ab]/(ab*ab)
=5-2ab+2ab/(ab*ab)+(a-b)(a-b)/(ab*ab)
=5-2ab+2/ab+(a-b)(a-b)/(ab*ab)
ab<=1/4 所以-2ab>=-1/2 2/ab>=8 (a-b)(a-b)/(ab*ab)>=0
（a+1/a)^2+(b+1/b)^2>=5-1/2+8+0=25/2
(当且仅当a=b=1/2时，取等号）

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由ab<=(a+b)^2/4=1/4
所以1/a+1/b+1/ab>=2*1/（1/4）>=8
（a+1/a)^2+(b+1/b)^2
=a^2+2+1/a^2+b^2+2+1/b^2
=(a^2+b^2+2ab)+1/a^2+1/b^2+4-2ab
=(a+b)^2+1/a^2+1/b^2+4-2ab
=1+1/a^2+1/b^2+4-2ab
>=1+2/ab+4-2*(1/4)=
>=1+2*4+4-2*(1/4)=25/2

如图所示
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