Question

The maximum value of \( x^4 y^4 \) when \( a^2 x^4 + b^2 y^4 = c^6 \) is:

• A. \( \dfrac{c^{12}}{16 a^4 b^4} \)
• B. \( \dfrac{c^{12}}{4 a^2 b^2} \)
• C. \( \dfrac{c^6}{(a + b)^{12}} \)
• D. \( \dfrac{c^6}{a^4 + b^4} \)

Hint:

Optimization under a constraint often uses Lagrange multipliers or symmetry. Use algebraic substitution to simplify the constraint.

Answer 1

The Correct Option is B

Using the method of Lagrange multipliers or the AM-GM inequality under the given constraint, we can find that the maximum value occurs when \[ a^2 x^4 = b^2 y^4 \Rightarrow \frac{x^4}{b^2} = \frac{y^4}{a^2} \] Solving this system under the constraint \( a^2 x^4 + b^2 y^4 = c^6 \) gives the result: \[ x^4 y^4 = \frac{c^{12}}{4 a^2 b^2} \]