證明函數(shù)f（x）=（x2+1）/（x4+1）在定義域R內(nèi)有界

數(shù)學(xué)人氣：892 ℃時(shí)間：2019-09-18 01:47:10

優(yōu)質(zhì)解答

定義域?yàn)镽,
令t=x^2>=0
則f=(t+1)/(t^2+1)=t/(t^2+1)+1/(t^2+1)
t=0時(shí),f=1
t>0時(shí),f=1/(t+1/t)+1/(t^2+1)
因?yàn)閠+1/t>=2,故0∣f(x)∣=∣(x^2+1)/(x^4+1)∣≤(x^2+1)^2/(x^4+1）=(x^4+1+2x^2)/(x^4+1)=1+2x^2/(x^4+1)≤1+1=2故f(x)在R內(nèi)有界書(shū)上是這樣寫(xiě)的，1+2x^2/(x^4+1)≤1+1 這一步?jīng)]看懂這是用公式：2ab<=a^2+b^2a=x^2, b=1因此有：2x^2<=x^4+12x^2/(x^4+1)≤11+2x^2/(x^4+1)≤1+1