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Given $ab+bc+ca =12$ then what is the least value of $a+b+c$ can we use AM - GM inequality for solving this problem

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    Please detail out exactly what everything means, and what you have tried, and where you get stuck. Also, this reads like a *command*. We prefer to be *asked* to solve things. – 2017-01-19
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    a=2 b=2 c=2 so a+b+c = 6 – 2017-01-19
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    I tried solving it by using AM - GM inequality and used triangle inequality – 2017-01-19

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$(a+b+c)^2\geq3(ab+ac+bc)=36$, which gives $a+b+c\geq6$.

The equality occurs for $a=b=c=2$.

We can prove $(a+b+c)^2\geq3(ab+ac+bc)$ by AM-GM: $$\sum_{cyc}(a^2+b^2)\geq2\sum_{cyc}ab$$ or $$\sum_{cyc}(a^2-ab)\geq0$$ or $$(a+b+c)^2\geq3(ab+ac+bc)$$