Question

已知函数y=log3（(mx^2+8x+n)/(x^2+1)的定义域为R，值域为(0<=y<=2),求m，n的值

谢谢啊，很急

Answer 1

mx^2+8x+n >0的解为x∈R(显然m≠0)

m>0
8² -4mn>=0 (1)

m < 0那是不可能的

0<=log3（(mx^2+8x+n)/(x^2+1))<=2

即1 <=（(mx^2+8x+n)/(x^2+1))<= 9

化简后,

(m-1)x² + 8x +(n-1)>=0 (x∈R) -----> m>1,8²-4(m-1)(n-1)>=0 (2)

(m-9)x² + 8x + (n -9)<=0 (x∈R) ----->m <9,8²-4(m-9)(n-9)<=0 (3)

由(1),(2),(3)得1<m<9 ,

(m-1)(n-1)=16,(m-9)(n-9)=16(判别式等于零才有最值)

所以m =5, n=5