Question

The minimum value of $ax+by$ where $xy=c^{2}$ is}

• A. $2c\sqrt{ab}$
• B. $2ab\sqrt{c}$
• C. $-2c\sqrt{ab}$
• D. $2c(ab)$

Hint:

AM-GM is often the fastest way to find minimums for sums with a constant product.

Answer 1

The Correct Option is A

Step 1: Concept
Use the AM-GM inequality: For positive numbers, $\frac{A+B}{2} \ge \sqrt{AB}$.

Step 2: Analysis
- Let the two numbers be $ax$ and $by$. - $\frac{ax + by}{2} \ge \sqrt{(ax)(by)} = \sqrt{ab(xy)}$.

Step 3: Calculation
- Given $xy = c^2$, the RHS becomes $\sqrt{abc^2} = c\sqrt{ab}$. - So, $ax + by \ge 2c\sqrt{ab}$.

Step 4: Conclusion
Minimum value is $2c\sqrt{ab}$. Final Answer: (A)