b) a + b + c = 1 \rightarrow 1-a=b+c; 1-b=c+a; 1-c=a+b(1 - a)(1 - b)(1 - c)=(a+b)(b+c)(c+a) \geq 2\sqrt{ab}*2\sqrt{bc}*2\sqrt{ca}=8abc (dpcm)c) Có 1+1/a=1+1/3a+1/3a+1/3a \geq 4*\sqrt[4]{\frac{1}{(3a)3}}cmtt: 1+1/b\geq 4*\sqrt[4]{\frac{1}{(3b)3}}1+1/c \geq 4*\sqrt[4]{\frac{1}{(3c)3}}\rightarrow (1+1/a)(1+1/b)(1+1/c) \geq 64*\sqrt[4]{\frac{1}{(27abc)3}} \geq 64*\sqrt[4]{(a+b+c)9}=64(dpcm)
b)
$a+b+c=1 \rightarrow 1-a=b+c; 1-b=c+a; 1-c=a+b
$$(1-a)(1-b)(1-c)=(a+b)(b+c)(c+a) \geq 2\sqrt{ab}2\sqrt{bc}2\sqrt{ca}=8abc
$ (dpcm)
c) Có
$1+
\frac{1
}{a
}=1+
\frac{1
}{3a
}+
\frac{1
}{3a
}+
\frac{1
}{3a
} \geq 4\sqrt[4]{\frac{1}{3a
}^3}
$cmtt:
$1+
\frac{1
}{b
}\geq4\sqrt[4]{\frac{1}{3b
}^3}
$ $1+
\frac{1
}{c
} \geq 4\sqrt[4]{\frac{1}{3c}
^3}
$$\rightarrow (1+
\frac{1
}{a
})(1+
\frac{1
}{b
})(1+
\frac{1
}{c
}) \geq 64\sqrt[4]{\frac{1}{27abc}
^3} \geq 64\sqrt[4]{(a+b+c)
^9}
$ (
đpcm)