Question

If a,b,c \in R^+ , and a+b+c=1 , prove 1/a+1/b+1/c\geq 9

Collected in the board: Inequality

Steven Zheng posted 2 months ago


Answer

\because a+b+c =1

\dfrac{1}{a} +\dfrac{1}{b}+\dfrac{1}{c}

=\dfrac{a+b+c}{a}+\dfrac{a+b+c}{b}+\dfrac{a+b+c}{c}

=3+\dfrac{b}{a}+\dfrac{a}{b}+\dfrac{a}{c}+\dfrac{c}{a}+\dfrac{b}{c}+\dfrac{c}{b}

\geq 3+2+2+2=9

Only if a=b=c , two sides of the inequality are equal.

Steven Zheng posted 2 months ago

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