To find the smallest possible value of the expression

$\frac{32}{a} + \frac{128}{b} + ab$

where $a$ and $b$ are positive real numbers, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

The AM-GM inequality states that for non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. For three terms, it can be applied as follows:

$\frac{32}{a} + \frac{128}{b} + ab \geq 3 \sqrt[3]{\frac{32}{a} \cdot \frac{128}{b} \cdot ab}$

1. Compute the product under the cube root: $\frac{32}{a} \cdot \frac{128}{b} \cdot ab = 32 \cdot 128 = 4096$

2. Take the cube root: $\sqrt[3]{4096} = 16$

3. Apply the AM-GM inequality: $\frac{32}{a} + \frac{128}{b} + ab \geq 3 \cdot 16 = 48$

Thus, the minimum possible value of $\frac{32}{a} + \frac{128}{b} + ab$ is $48$.

Conclusion

The smallest possible value of the expression is $\boxed{48}$, which matches option 4.

Would you like further clarification on any step?

Related Questions

1. How does the AM-GM inequality help in optimization problems?
2. Can this minimum be achieved for specific values of $a$ and $b$?
3. Why is the cube root used in this AM-GM application?
4. How would the solution change if the expression involved different constants?
5. Could the AM-GM inequality be applied to more complex expressions?

Tip

The AM-GM inequality is a powerful tool for minimizing or maximizing expressions involving products and sums, especially when symmetry is present.