Question

若x∈R,求x*2+1/(x*+2)的最小值

Answer 1

x^2+1/(x^2+2)
＝(x^2+2)+1/(x^2+2)-2.
设t＝x^2+2≥2,
构造对勾函数f(t)＝t+1/t,
t∈(1,+∞)时,f(t)单调递增.
∴f(t)≥f(2)＝2+1/2＝5/2.
故t＝x^2+2＝2,即x＝0时,
所求最小值为：5/2-2＝1/2.