<s>[INST] If x,y,z is positive then find the minimum value of $x+y+z+\frac{1}{x} + \frac{1}{y} +\frac{1}{z}+1$ If x,y,z is positive then find the minimum value of $x+y+z+\frac{1}{x} + \frac{1}{y} +\frac{1}{z}+1$
I know this can be solved by the relation AM$\geq$GM$\geq$HM. But I do not how to apply here and proceed.
Please help me out. Thank You!
 [/INST] $x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+1 \geq 6 \sqrt[6]{x \cdot y \cdot z \cdot \frac{1}{x} \cdot \frac{1}{y} \cdot \frac{1}{z} } + 1=7$, where the equality is obtained in $x = y = z = \frac{1}{x} = \frac{1}{y} = \frac{1}{z} =1.$
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