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: Understanding the Question:
We need to find the minimum of a linear combination of $x$ and $y$ subject to a constant product (rectangular hyperbola constraint).

Step 2: Key Formula or Approach:
For positive numbers, the AM-GM inequality is often easiest: \[ \frac{ax + by}{2} \ge \sqrt{ax \cdot by} \]

Step 3: Detailed Explanation:
Assume $a, b, x, y$ are positive.
According to Arithmetic Mean $\ge$ Geometric Mean:
$\frac{ax + by}{2} \ge \sqrt{(ax)(by)}$
$\frac{ax + by}{2} \ge \sqrt{ab(xy)}$
Substitute $xy = c^2$:
$\frac{ax + by}{2} \ge \sqrt{ab \cdot c^2}$
$\frac{ax + by}{2} \ge c\sqrt{ab}$
$ax + by \ge 2c\sqrt{ab}$.
The minimum value is attained when $ax = by$.

Step 4: Final Answer:
The minimum value is $2c\sqrt{ab}$.