Optimizing the Expression a * b / abc Given a * b * c 1

When dealing with expressions involving variables and optimizing them under certain constraints, the goal is often to find the minimum or maximum possible value. In this case, we are considering the expression (frac{a b}{abc}) under the constraint (a b c 1), where (a, b,) and (c) are non-negative.

Step-by-Step Solution

To solve this problem, we start by expressing (c) in terms of (a) and (b).

(c 1 - a - b)

Next, we rewrite the expression we want to minimize:

(frac{a b}{abc} frac{a b}{ab(1 - a - b)} frac{1}{1 - a - b})

Transformation and Simplification

We introduce a variable (x ab). Then, we express (c) in terms of (x), giving us:

(c 1 - x)

Our expression now becomes:

(frac{1}{1 - x})

To minimize the expression, we need to find the maximum value of (1 - x). This leads us to express (ab) in terms of (x).

By the AM-GM inequality, the maximum value of (ab) occurs when (a b). Therefore, we set (a b frac{x}{2}).

(ab left(frac{x}{2}right)left(frac{x}{2}right) frac{x^2}{4})

Substituting this back into our expression:

(frac{1}{1 - frac{x^2}{4}} frac{4}{x^2 - 4x 4})

Minimizing the Expression

We need to minimize (frac{4}{x^2 - 4x 4}). We can denote (f(x) x^2 - 4x 4). To find the maximum of this function, we take its derivative:

(f'(x) 2x - 4)

Solving for (x) where (f'(x) 0), we get:

(2x - 4 0 Rightarrow x 2)

However, since (x ab) and (a, b geq 0), the correct range for (x) is (0 leq x leq 1). Within this range, the function (f(x)) is maximized at:

(x frac{1}{2})

At this point:

(fleft(frac{1}{2}right) left(frac{1}{2}right)^2 - 4left(frac{1}{2}right) 4 frac{1}{4})

Thus, the minimum possible value is:

(frac{4}{frac{1}{4}} 16)

Conclusion

The minimum possible value of (frac{a b}{abc}) when (a b c 1) and (a, b, c geq 0) is (16). This value is achieved when (a b frac{1}{4}) and (c frac{1}{2}).

Final Answer: The minimum possible value is (boxed{16}).