According to A.M G.M inequality:
$(a+b)/2\geq\sqrt{ab}$, for all $a,b>0$.
The equality holds when $(a = b)$, and when this condition is satisfied, the LHS expression attains its minimum value. Does this mean that the sum of two positive numbers is at its minimum when they are equal? What have I misunderstood here?
If you know that $ab=c$ where $c$ is a constant, then, in this case, $a+b$ will be minimised when $a=b=\sqrt{c}$.
If you know that $a+b=c$ where $c$ is a constant, then, in this case, $ab$ will be maximised when $a=b=c/2$.
If you don't know any of these, you cannot use the word "minimised" or "maximised" because these words are used with respect to a constant. All the statement tells us otherwise is that $\dfrac{a+b}{2}\geq\sqrt{ab}$. That's all.