Question

1. Cho a,b >0; a+b ≤ 1

Tìm min \(N=ab+\dfrac{1}{ab}\)

2. Cho a,b,c >0 t/m: a+b+c ≥ 6

Tìm min \(P=5a+6b+7c+\dfrac{1}{a}+\dfrac{8}{b}+\dfrac{27}{c}\)

3. Cho a,b,c ∈ \(\left[-1;2\right]\) và \(a^2+b^2+c^2=6\)

\(CM:\) a+b+c ≥ 0

Answer 1

Câu 1

\(a+b\ge2\sqrt{ab}\Leftrightarrow ab\le\dfrac{\left(a+b\right)^2}{4}\\ \Leftrightarrow N=ab+\dfrac{1}{16ab}+\dfrac{15}{16ab}\ge2\sqrt{\dfrac{1}{16}}+\dfrac{15}{4\left(a+b\right)^2}\ge\dfrac{1}{2}+\dfrac{15}{4}=\dfrac{17}{4}\)

Dấu \("="\Leftrightarrow a=b=\dfrac{1}{2}\)

Câu 2:

\(P=a+\dfrac{1}{a}+2b+\dfrac{8}{b}+3c+\dfrac{27}{c}+4\left(a+b+c\right)\\ P\ge2\sqrt{1}+2\sqrt{16}+2\sqrt{81}+4\cdot6=2+8+18+4=32\)

Dấu \("="\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=2\\c=3\end{matrix}\right.\)

Câu 3: Cho a,b,c là các số thuộc đoạn [ -1;2 ] thõa mãn \(a^2+b^2+c^2=6.\) CMR : \(a+b+c>0\) - Hoc24