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Áp dụng BĐT Cauchy
\(1\ge a+b\ge2\sqrt{ab}\Leftrightarrow1\ge4ab\Leftrightarrow ab\le\dfrac{1}{4}\)
Ta có:
\(A=ab+\dfrac{1}{ab}=ab+\dfrac{1}{16ab}+\dfrac{15}{16ab}\)
\(\ge2.\sqrt{ab.\dfrac{1}{16ab}}+\dfrac{15}{16.\dfrac{1}{4}}=\dfrac{17}{4}\)
Vậy \(A_{MIN}=\dfrac{17}{4}\) khi và chỉ khi \(a=b=\dfrac{1}{2}\)
\(P=\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}+\dfrac{1}{a^2+b^2+c^2}\ge\dfrac{\left(1+1+1\right)^2}{ab+bc+ca}+\dfrac{1}{a^2+b^2+c^2}\) (BĐT Cauchy Schwarz)
\(=\dfrac{9}{ab+bc+ca}+\dfrac{1}{a^2+b^2+c^2}\)
\(=\dfrac{1}{ab+bc+ca}+\dfrac{1}{ab+bc+ca}+\dfrac{1}{a^2+b^2+c^2}+\dfrac{7}{ab+bc+ca}\)
\(\ge\dfrac{\left(1+1+1\right)^2}{a^2+b^2+c^2+2ab+2ac+2bc}+\dfrac{7}{ab+bc+ca}\)
\(=\dfrac{9}{\left(a+b+c\right)^2}+\dfrac{7}{ab+bc+ca}\)
Ta có: \(ab+bc+ca\le\dfrac{\left(a+b+c\right)^2}{3}=\dfrac{1}{3}\) .Thế vào biểu thức
\(\Rightarrow P\ge9+\dfrac{7}{\dfrac{1}{3}}=9+21=30\)
\(\Rightarrow P_{min}=30\) khi \(a=b=c=\dfrac{1}{3}\)
\(S=\dfrac{1}{a^3+b^3}+\dfrac{\dfrac{9}{4}}{3a^2b}+\dfrac{\dfrac{9}{4}}{3ab^2}+\dfrac{1}{4ab}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\)
Áp dụng bđt Cauchy-Schwarz dạng Engel có:
\(S\ge\dfrac{\left(1+\dfrac{3}{2}+\dfrac{3}{2}\right)^2}{a^3+3a^2b+3ab^2+b^3}+\dfrac{1}{4ab}.\dfrac{4}{a+b}\)
\(\Leftrightarrow S\ge\dfrac{16}{\left(a+b\right)^3}+\dfrac{1}{\left(a+b\right)^2}.\dfrac{4}{a+b}\)
\(\Leftrightarrow S\ge\dfrac{16}{1}+\dfrac{1}{1}.\dfrac{4}{1}=20\)
Dấu "=" xảy ra khi \(a=b=\dfrac{1}{2}\)
Vậy GTNN của \(S=20\) khi \(a=b=\dfrac{1}{2}\)
\(A=\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}+\dfrac{1}{4xy}+4xy+\dfrac{5}{4xy}\)
\(\ge\dfrac{4}{x^2+y^2+2xy}+2\sqrt{\dfrac{1}{4xy}.4xy}+\dfrac{5}{4.\dfrac{\left(x+y\right)^2}{4}}\)
\(\ge\dfrac{4}{1^2}+2+\dfrac{5}{1^2}\) (do \(x+y\le1\))
\(=11\)
Dấu "=" xảy ra \(\Leftrightarrow x=y=\dfrac{1}{2}\)
Vậy GTNN của A là 11.
\(A=a^2+b^2+\dfrac{1}{a^2}+\dfrac{1}{b^2}\)
\(A=a^2+\dfrac{1}{16a^2}+b^2+\dfrac{1}{16b^2}+\dfrac{15}{16}\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}\right)\)
\(A\ge2\sqrt{a^2\cdot\dfrac{1}{16a^2}}+2\sqrt{b^2\cdot\dfrac{1}{16b^2}}+\dfrac{15}{16}\cdot2\cdot\sqrt{\dfrac{1}{a^2b^2}}\)
\(A\ge1+\dfrac{15}{8ab}\ge1+\dfrac{15}{2\left(a+b\right)^2}\ge\dfrac{17}{2}\)
"="<=>x=y=0,5
\(A=\dfrac{1}{a^2+b^2}+\dfrac{1}{ab}+4ab=\dfrac{1}{a^2+b^2}+\dfrac{1}{2ab}+\dfrac{1}{2ab}+8ab-4ab\ge\dfrac{4}{a^2+b^2+2ab}+2\sqrt{\dfrac{1}{2}.8}-\dfrac{4.\left(a+b\right)^2}{4}=\dfrac{4}{\left(a+b\right)^2}+4-\left(a+b\right)^2\ge4+4-1=7\Rightarrow minA=7\Leftrightarrow a=b=\dfrac{1}{2}\)