Ta có:

\(M=\frac{19a+3}{1+b^2}+\frac{19b+3}{c^2+1}+\frac{19c+3}{a^2+1}\)

\(=19a-\frac{19ab^2-3}{b^2+1}+19b-\frac{19bc^2-3}{c^2+1}+\frac{19ca^2-3}{a^2+1}\)

\(\ge19\left(a+b+c\right)-\frac{19ab^2-3}{2b}-\frac{19bc^2-3}{2c}-\frac{19ca^2-3}{2a}\)

\(=19\left(a+b+c\right)-19\left(\frac{ab}{2}+\frac{bc}{2}+\frac{ca}{2}\right)+\frac{3}{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

\(\ge19.3-\frac{19.3}{2}+\frac{3}{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{19.3}{2}+\frac{3}{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

Lại có:

\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2\ge3\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)\ge3\frac{\left(1+1+1\right)^2}{ab+bc+ca}=\frac{3.9}{3}=9\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge3\)

\(\Rightarrow M\ge\frac{19.3}{2}+\frac{3}{2}.3=33\)

\(\)

a/ Đặt: \(x+\frac{1}{x}=a\)

Ta có: \(x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)^3-3\left(x+\frac{1}{x}\right)=a^3-3a\)

\(x^6+\frac{1}{x^6}=\left(x^3+\frac{1}{x^3}\right)^2-2=\left(\left(x+\frac{1}{x}\right)^3-3\left(x+\frac{1}{x}\right)\right)^2-2\)

\(=\left(a^3-3a\right)^2-2\)

\(\Rightarrow M=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+x^3+\frac{1}{x^3}}\)

\(=\frac{a^6-\left(a^3-3a\right)^2+2-2}{a^3+a^3-3a}\)

\(=\frac{\left(a^3+a^3-3a\right)\left(a^3-a^3+3a\right)}{\left(a^3+a^3-3a\right)}=3a\)

\(=3.\left(x+\frac{1}{x}\right)=\frac{3x^2+3}{x}\)

b/ \(\frac{3x^2+3}{x}=3x+\frac{3}{x}\ge2.3=6\)

Đấu =  xảy ra khi \(x=\frac{1}{x}\Leftrightarrow x=1\)

\(A=\frac{8a^2+b}{4a}+b^2=2a+\frac{b}{4a}+b^2=\left(b^2+\frac{b}{4a}+\frac{a}{2}\right)+\frac{3}{2}a\)

\(\ge3\sqrt[3]{b^2.\frac{b}{4a}.\frac{a}{2}}+\frac{3}{2}a=\frac{3}{2}a+\frac{3}{2}b=\frac{3}{2}\left(a+b\right)\ge\frac{3}{2}\) 

Dấu "=" xảy ra \(\Leftrightarrow a=b=\frac{1}{2}\)

\(\frac{a}{9b^2+1}=\frac{a\left(9b^2+1\right)-9ab^2}{9b^2+1}=a-\frac{9ab^2}{9b^2+1}\ge a-\frac{9ab^2}{2\sqrt{9b^2.1}}=\)

\(=a-\frac{9ab^2}{6b}=a-\frac{3ab}{2}\)

Tương tự với các biểu thức còn lại, kết hợp với 

\(ab+bc+ca\le\frac{1}{3}\left(a+b+c\right)^2\)

là được đáp án.

\(A=a^4-2a^3+3a^2-4a+5\)

\(=\left(a^4-2a^3+a^2\right)+\left(2a^2-4a+2\right)+3\)

\(=\left(a^2-a\right)^2+2\left(a-1\right)^2+3\ge3\)

Dấu =  xảy ra khi a = 1

1 nha bạn

\(A=a^2+b^2+\frac{1}{a^2}+\frac{1}{b^2}\)

\(=a^2+b^2+\frac{b^2+a^2}{a^2b^2}\ge0\)

\(MinA=0\Leftrightarrow\hept{\begin{cases}a^2=0\\b^2=0\end{cases}\Rightarrow\hept{\begin{cases}a=0\\b=0\end{cases}}}\)

bạn lm sai rồi

Từ giả thuyết ta có \(\hept{\begin{cases}\frac{1}{4}\ge ab\\ab+\frac{1}{16ab}\ge\frac{1}{2}\end{cases}}\)

Ta có \(\frac{1}{a^2}+\frac{1}{b^2}+a^2+b^2\)\(\ge\)\(2\left(ab+\frac{1}{ab}\right)\)

= \(2\left(ab+\frac{1}{16ab}+\frac{15}{16ab}\right)\)

\(\ge\)\(2\left(\frac{1}{2}+\frac{15}{16×\frac{1}{4}}\right)\)= \(\frac{17}{2}\)

Vậy GTNN là \(\frac{17}{2}\)đạt được khi a = b = \(\frac{1}{2}\)