Question

If \( x+y=50 \), then the maximum value of \( \sqrt{4xy} \) is

• A. \( 25 \)
• B. \( 50 \)
• C. \( 100 \)
• D. \( 625 \)
• E. \( 2500 \)

Hint:

Whenever the sum of two numbers is fixed, their product is maximum when the two numbers are equal. This is a direct application of AM-GM inequality.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to find the maximum value of an expression involving two variables, \( x \) and \( y \), subject to a linear constraint. This can be solved using calculus (by substitution) or by using the Arithmetic Mean-Geometric Mean (AM-GM) inequality. The AM-GM inequality provides a very quick solution for this type of problem.
Step 2: Key Formula or Approach:
The AM-GM inequality states that for non-negative numbers \( x \) and \( y \), the arithmetic mean is always greater than or equal to the geometric mean:
\[ \frac{x+y}{2} \ge \sqrt{xy} \] Equality holds if and only if \( x = y \).
Step 3: Detailed Explanation:
We are given the constraint \( x + y = 50 \). We assume \( x \) and \( y \) are non-negative for the product to be maximized.
Applying the AM-GM inequality:
\[ \frac{x+y}{2} \ge \sqrt{xy} \] Substitute the given value of \( x+y \):
\[ \frac{50}{2} \ge \sqrt{xy} \] \[ 25 \ge \sqrt{xy} \] This tells us that the maximum value of \( \sqrt{xy} \) is 25.
The expression we need to maximize is \( \sqrt{4xy} \). We can simplify this expression:
\[ \sqrt{4xy} = \sqrt{4} \cdot \sqrt{xy} = 2\sqrt{xy} \] Now, we can find the maximum value of this expression:
\[ \text{Max}(\sqrt{4xy}) = 2 \cdot \text{Max}(\sqrt{xy}) \] \[ \text{Max}(\sqrt{4xy}) = 2 \cdot 25 = 50 \] This maximum value occurs when \( x=y \). From the constraint \( x+y=50 \), this happens when \( x=y=25 \).
Step 4: Final Answer:
The maximum value of \( \sqrt{4xy} \) is 50. This corresponds to option (B).