Question

If \[ a^2 x^4 + b^2 y^4 = c^6, \] then the maximum value of \( xy \) is:

• A. \( \frac{c^3}{2ab} \)
• B. \( \frac{c^3}{\sqrt{2ab}} \)
• C. \( \frac{c^3}{ab} \)
• D. \( \frac{c^3}{\sqrt{ab}} \)

Hint:

To find maximum values in constrained equations, use AM-GM or Lagrange multipliers.

Answer 1

The Correct Option is B

Step 1: Using AM-GM inequality For maximum \( xy \), apply the method of Lagrange multipliers or use the AM-GM inequality: \[ a^2 x^4 + b^2 y^4 \geq 2 \sqrt{a^2 x^4 b^2 y^4}. \] Substituting: \[ c^6 \geq 2 \sqrt{a^2 x^4 b^2 y^4}. \] Solving for \( xy \): \[ xy \leq \frac{c^3}{\sqrt{2ab}}. \]