Question

已知函數(shù)f(x)=log_ax(a>0且a≠1),(x∈(0,+∞)),若x₁,x₂∈(0,+∞),判斷［f(x₁)+f(x₂)］與f()的大小，并加以證明. 

Answer 1

［f(x₁)+f(x₂)］≥f()(當(dāng)且僅當(dāng)x₁=x₂時(shí)取“=”號(hào)）.

f(x₁)+f(x₂)=log_ax₁+log_ax₂=log_ax₁x₂,
∵x₁,x₂∈(0,+∞),x₁x₂≤()²(當(dāng)且僅當(dāng)x₁=x₂時(shí)取“=”號(hào))，
當(dāng)a>1時(shí)，有l(wèi)og_ax₁x₂≤log_a()²,
∴log_ax₁x₂≤log_a()，(log_ax₁+log_ax₂)≤log_a,
即f(x₁)+f(x₂)］≤f()(當(dāng)且僅當(dāng)x₁=x₂時(shí)取“=”號(hào))
當(dāng)0＜a＜1時(shí)，有l(wèi)og_ax₁x₂≥log_a()²,
∴(log_ax₁+log_ax₂)≥log_a,即［f(x₁)+f(x₂)］≥f()(當(dāng)且僅當(dāng)x₁=x₂時(shí)取“=”號(hào)）.