已知f(x)=(x^4+kx^2+1)/(x^4+x^2+1)(k是实常数),则f(x)的最大值与最小值的乘积为?

Question

已知f(x)=(x^4+kx^2+1)/(x^4+x^2+1)(k是实常数),则 f(x)的最大值与最小值的乘积为?

Answer 1

f(x)=(x^4+kx^2+1)/(x^4+x^2+1)=1+[(k-1)x^2/(x^4+x^2+1)]
=1+[(k-1)/(x^2+(1/x^2)+1)]
当k>1时,(x^2+(1/x^2)+1)>=2√(x^2*(1/x^2))+1=3,
所以f(x)1, 所以最小值为1.
乘积为(k+2)/3.
同理,k