Question

函数Y=X^2+1/X^2-X-1/X的值域为?

Answer 1

Y=(X+1/X)^2-(X+1/X)-2=(X+1/X-1/2)^2-2-1/4=(X+1/X-1/2)^2-9/4

X+1/X≥2或≤-2

X+1/X-1/2≥2-1/2=3/2或X+1/X-1/2≤-2-1/2=-5/2

(X+1/X-1/2)^2≥9/4或(X+1/X-1/2)^2≥25/4

综上(X+1/X-1/2)^2≥9/4

所以Y≥9/4-9/4=0