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来源：学生作业帮助网 编辑：作业帮 时间：2024/05/04 13:11:02
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[1/x^2+1/(a-x)^2]•a^2
=[1/x^2+1/(a-x)^2]•[x+(a-x)]^2
=[1/x^2+1/(a-x)^2]•[x^2+2x(a-x)+(a-x)^2]
=1+2x(a-x)/x^2+(a-x)^2/x^2+ x^2/(a-x)^2+2x(a-x) /(a-x)^2+1
=1+2(a-x)/x+(a-x)^2/x^2+ x^2/(a-x)^2+2 x /(a-x) +1
=2+[2(a-x)/x+2 x /(a-x) ]+[(a-x)^2/x^2+ x^2/(a-x)^2]
≥2+2√[2(a-x)/x•2 x /(a-x) ]+ 2√[(a-x)^2/x^2•x^2/(a-x)^2]
=2+4+2=8,
∴1/x^2+1/(a-x)^2≥8/ a^2,
即1/x^2+1/(a-x)^2的最小值是8/ a^2,
不等式1/x^2+1/(a-x)^2≥2恒成立,
则有8/ a^2≥2,
∴0
[1/x^2+1/(a-x)^2]•a^2
=[1/x^2+1/(a-x)^2]•[x+(a-x)]^2
=[1/x^2+1/(a-x)^2]•[x^2+2x(a-x)+(a-x)^2]
=1+2x(a-x)/x^2+(a-x)^2/x^2+ x^2/(a-x)^2+2x(a-x) /(a-x)^2+1
=1+2(a-x)/x+(...
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[1/x^2+1/(a-x)^2]•a^2
=[1/x^2+1/(a-x)^2]•[x+(a-x)]^2
=[1/x^2+1/(a-x)^2]•[x^2+2x(a-x)+(a-x)^2]
=1+2x(a-x)/x^2+(a-x)^2/x^2+ x^2/(a-x)^2+2x(a-x) /(a-x)^2+1
=1+2(a-x)/x+(a-x)^2/x^2+ x^2/(a-x)^2+2 x /(a-x) +1
=2+[2(a-x)/x+2 x /(a-x) ]+[(a-x)^2/x^2+ x^2/(a-x)^2]
≥2+2√[2(a-x)/x•2 x /(a-x) ]+ 2√[(a-x)^2/x^2•x^2/(a-x)^2]
=2+4+2=8，
∴1/x^2+1/(a-x)^2≥8/ a^2,
即1/x^2+1/(a-x)^2的最小值是8/ a^2,
不等式1/x^2+1/(a-x)^2≥2恒成立,
则有8/ a^2≥2，
∴0
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