解:
a^2(cotx)^2+2abtanx
=a^2(cotx)^2+abtanx+abtanx
>=3(a^4b^2)^(1/3) 。。。。。。(1)
同理b^2(tanx)^2+2ab>=3(a^2b^4)^(1/3) 。 。。。。。(2)
由(1)+(2),得
a^2(cotx)^2+b^2(tanx)^2+2abtanx+2abtanx>=3(a^4b^2)^(1/3)+3(a^2b^4)^(1/3)
--->(a/sinx)^2+(b/cosx)^2+(2ab/sinxcosx)>=a^2+b^2+3(a^4b^2)^(1/3)+3(a^2b^4)^(1/3)
--->(a/...全部
解:
a^2(cotx)^2+2abtanx
=a^2(cotx)^2+abtanx+abtanx
>=3(a^4b^2)^(1/3) 。。。。。。(1)
同理b^2(tanx)^2+2ab>=3(a^2b^4)^(1/3) 。
。。。。。(2)
由(1)+(2),得
a^2(cotx)^2+b^2(tanx)^2+2abtanx+2abtanx>=3(a^4b^2)^(1/3)+3(a^2b^4)^(1/3)
--->(a/sinx)^2+(b/cosx)^2+(2ab/sinxcosx)>=a^2+b^2+3(a^4b^2)^(1/3)+3(a^2b^4)^(1/3)
--->(a/sinx+b/cosx)^2>=[a^(2/3)+b^(2/3)]^3
--->a/sinx+b/cosx>=[a^(2/3)+b^(2/3)]^(3/2)
即所求最小值为[a^(2/3)+b^(2/3)]^(3/2)。
。收起