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. $\frac{c^2}{2\sqrt{ab}}$
• C. $\frac{c}{2\sqrt{ab}}$
• D. $\frac{c^2}{\sqrt{2ab}}$

Answer 1

Answer: (D)

If the sum of two positive quantities is a constant,
then their product is maximum, when they are equal.
$\therefore a^2{x}^4\cdot b^2{y}^2$ is maximum when
$a^2{x}^4=b^2{y}^4=\frac{1}{2}\left(a^2{x}^4=\frac{1}{2}\left(a^2{x}^4+b^2{y}^4\right)\right)$
$=\frac{c^4}{2}$
$\therefore$ maximum value of
$a^2{x}^4\cdot b^2{y}^4=\frac{c^4}{2}\cdot \frac{c^4}{2}=\frac{c^8}{4}$
Maximum value of
$xy={\left(\frac{c^8}{4a^2{b}^2}\right)}^{1/4}=\frac{c^2}{\sqrt{2ab}}$