Question

已知函数f(x)=log3(ax^2+8x+b)/(x^2+1)的定义域为（-∞,+∞）,值域为[0,2],求a,b的值

Answer 1

f(x)=log3(ax^2+8x+b)/(x^2+1)值域为[0,2],

则y=ax^2+8x+b)/(x^2+1)值域为[1,9],

y=ax^2+8x+b)/(x^2+1)去分母得(a-y)x^2+8x+b-y=0,

由△=64-4(a-y)(b-y)≥0得

y^2-(a+b)y+ab-16=0,

所以1,9是y^2-(a+b)y+ab-16=0的根,

a+b=10,ab-16=9,a=b=5.