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I don't know if this inequality has a name. But in theory this is taught after the AM ≥ GM, then followed by saying that sum of a positive number and its reciprocal is always greater than or equal to 2.
I dont think there is any name for this equation.But the proof is very simple.Just differentiate the LHS of the equation and equate it to zero.This gives x=1 for extreme value(minimum in this case).Then substitiuting x=2 yields x+1/x=2.i.e minimum value of the expression is 2.