柯西不等式：∑(i=1,n)|ai×∑(i=1,n)|bi≥∑(i=1,n)|(aibi),即：
(a1²+a2²+a3²+···+an²)(b1²+b2²+b3²+···+bn²)≥(a1b1+a2b2+a3b3+···+anbn)²
此題不能直接或主要運(yùn)用柯柯西不等式!
f(x)=√(x-3)+√(6-x)
先確定f(x)的定義域：
x-3≥0,6-x≥；解得3≤x≤6.
f(x)=√(x-3)+√(6-x)
=√(√(x-3)+√(6-x))²
=√(√(x-3)²+√(6-x)²+2√((x-3)(6-x)))
=√((x-3+6-x)+2√(-x²+9x-18))
=√(3+2√((9/4)-(x-(9/2))²))
∵3≤x≤6
∴0≤√((9/4)-(x-(9/2))²)≤√(9/4)=3/2
∴√3≤√(3+2√((9/4)-(x-(9/2))²))≤√(3+2×(3/2))=√6
∴√3≤f(x)≤√6
當(dāng)x=3或6時(shí),f(x)min=√3；當(dāng)x=9/2時(shí),f(x)max=√6.
∴要f(x)√3即可!