a: \(a^2+\frac14\ge2\cdot\sqrt{a^2\cdot\frac14}=2\cdot\frac12\cdot a=a\)
\(b^2+\frac14\ge2\cdot\sqrt{b^2\cdot\frac14}=2\cdot b\cdot\frac12=b\)
\(c^2+\frac14\ge2\cdot\sqrt{c^2\cdot\frac14}=2\cdot c\cdot\frac12=c\)
Do đó: \(a^2+\frac14+b^2+\frac14+c^2+\frac14\ge a+b+c\)
=>\(a^2+b^2+c^2\ge\frac32-\frac34=\frac34\)
d: \(a+1\ge2\cdot\sqrt{a}\)
\(b+1\ge2\cdot\sqrt{b}\)
\(c+1\ge2\cdot\sqrt{c}\)
Do đó: \(\left(a+1\right)\left(b+1\right)\left(c+1\right)\ge2\cdot\sqrt{a}\cdot2\sqrt{b}\cdot2\sqrt{c}=8\sqrt{abc}=8\)
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