Question

Chứng minh bất đẳng thức: \(\sqrt{a+1}-\sqrt{a}< \dfrac{1}{2\sqrt{a}}\left(ĐK:a\ge1\right)\)

Answer 1

\(\Leftrightarrow2\sqrt{a\left(a+1\right)}-2a< 1\)

Lại có:\(2\sqrt{a\left(a+1\right)}\le a+a+1=2a+1\)

\(\Rightarrow2\sqrt{a\left(a+1\right)}-2a\le2a+1-2a=1\)

Dấu "=" không xảy ra

\(\Rightarrow\sqrt{a+1}-\sqrt{a}< \dfrac{1}{2\sqrt{a}}\)(đpcm)