命题函数y=a/cosx+b/sinx,(a、b∈R~+),x∈(0,1/2π)的最小值为(((a~2)~(1/3)+(b~2~(1/3))~3)~(1/2) 证明∵a~(1/3)cosx+b~(1/3)sinx ≤ ((a~2)~(1/3)+(b~2)~(1/3))~(1/2)(当且仅当x=arc tg(b/a)~(1/3)时等号成立), ∴((a~2)~(1/3)+(b~2)~(1/3))~3)~(1/2)y≥a~(1/3)cosx+b~3sinx)·(a/cosx+b/sinx)≥(a~(1/6)(cosx)~(1/2)(a/cosx)~(1/2)+b~(1/6)(sinx)~(1/2)·((b/sinx)~(1/2))~2=((a~2)~(1/3)+(b~2)~(1/3))~2(当且仅当x=arc tg(b/a)~(1/3)时等号成立),即 The proposition function y=a/cosx+b/sinx, (a, b∈R~+), the minimum value of x∈(0, 1/2π) is (((a~2)~(1/3)+( b~2~(1/3))~3)~(1/2) Prove that ~a~(1/3)cosx+b~(1/3)sinx≤((a~2)~(1/3 )+(b~2)~(1/3))~(1/2)(If and only if x=arc tg(b/a)~(1/3) holds the equal sign), ∴((a ~2)~(1/3)+(b~2)~(1/3))~3)~(1/2)y≥a~(1/3)cosx+b~3sinx)·(a/ Cosx+b/sinx)≥(a~(1/6)(cosx)~(1/2)(a/cosx)~(1/2)+b~(1/6)(sinx)~(1/ 2) ((b/sinx)~(1/2))~2=((a~2)~(1/3)+(b~2)~(1/3))~2(if and only When x=arc tg(b/a)~(1/3) the equal sign is established), ie