Question

If \( a^2 x^4 + b^2 y^4 = c^4 \), then the maximum value of \( xy \) is

• A. \( \frac{c}{\sqrt{ab}} \)
• B. \( \frac{c^2}{\sqrt{ab}} \)
• C. \( \frac{c^2}{2ab} \)
• D. \( \frac{c^2}{\sqrt{2ab}} \)

Hint:

When maximizing expressions subject to constraints, use methods like Lagrange multipliers or differentiate the function directly.

Answer 1

The Correct Option is D

Step 1: Use optimization to find the maximum value.
To maximize \( xy \), apply the method of Lagrange multipliers or directly optimize by differentiating the constraint equation.
Step 2: Conclusion.
The maximum value of \( xy \) is \( \frac{c^2}{\sqrt{2ab}} \). Final Answer: \[ \boxed{\frac{c^2}{\sqrt{2ab}}} \]