Question:
If $a^2 x^4 + b^2 y^4 = c^6$, then the maximum value of $xy$ is:
If $a^2 x^4 + b^2 y^4 = c^6$, then the maximum value of $xy$ is:
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To find maximum values in constrained equations, use AM-GM or Lagrange multipliers.
Updated On: Mar 18, 2026
- \( \frac{c^3}{2ab} \)
- \( \frac{c^3}{\sqrt{2ab}} \)
- \( \frac{c^3}{ab} \)
- \( \frac{c^3}{\sqrt{ab}} \)
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The Correct Option is B
Solution and Explanation
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 \( xy \):
\[
xy \leq \frac{c^3}{\sqrt{2ab}}.
\]
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