Question

1.Cho x\(\ge\)1 tìm Min P \(=3x+\frac{1}{2x}\)

2.Cho a\(\ge\)10;b\(\ge\)100;c\(\ge\)1000 tìm Min P \(=a+\frac{1}{a}+b+\frac{1}{b}+c+\frac{1}{c}\)

3. Cho a,b>0 CMR : \(\frac{a}{b}+\frac{b}{a}+\frac{8ab}{\left(a+b\right)^2}\ge4\)

Answer 1

1.

\(P=\frac{x}{2}+\frac{1}{2x}+\frac{5x}{2}\ge2\sqrt{\frac{x}{4x}}+\frac{5}{2}.1=\frac{7}{2}\)

Dấu "=" xảy ra khi \(x=1\)

2.

\(P=\frac{a}{100}+\frac{1}{a}+\frac{b}{10000}+\frac{1}{b}+\frac{c}{1000^2}+\frac{1}{c}+\frac{99}{100}a+\frac{9999}{10000}b+\frac{999999}{1000000}c\)

\(P\ge2\sqrt{\frac{a}{100a}}+2\sqrt{\frac{b}{10000b}}+2\sqrt{\frac{c}{1000000c}}+\frac{99}{100}.10+\frac{9999}{10000}.100+\frac{999999}{1000000}.1000=...\)

Bạn tự bấm máy tính

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}a=10\\b=100\\c=1000\end{matrix}\right.\)

3.

\(VT=\frac{a^2+b^2}{ab}+\frac{8ab}{\left(a+b\right)^2}\ge\frac{\left(a+b\right)^2}{2ab}+\frac{8ab}{\left(a+b\right)^2}\ge2\sqrt{\frac{8ab\left(a+b\right)^2}{2ab\left(a+b\right)^2}}=4\)

Dấu "=" xảy ra khi \(a=b\)