If $\log{\left( x^{3} + y^{3} \right) - \log_{10}{(x^{2} + y^{2} - xy) \leq 2}}$, then the maximum value of $xy$, for all $x \geq 0,\ y \geq 0$ is

Question

Dr. Abubacker · Accepted Answer

(a)
Given, $\log_{10}{\left( x^{3} + y^{3} ight) - \log_{10}{\left( x^{2} + y^{2} - xy ight) \leq 2}}$
$\Rightarrow \ \log_{10}{\frac{(x^{3} + y^{3})}{x^{2} + y^{2} - xy} \leq 2}$
$\Rightarrow \ \log_{10}{(x + y) \leq 2 \Rightarrow x + y \leq 100}$
Using AM $\geq$ GM
$	herefore\ \ \frac{x + y}{2} \geq \sqrt{xy}$
$\Rightarrow \ \ \ \sqrt{xy} \leq \frac{x + y}{2} \leq \frac{100}{2}$
$\Rightarrow \ \ \ xy \leq 2500$

Q. If $\log{\left( x^{3} + y^{3} \right) - \log_{10}{(x^{2} + y^{2} - xy) \leq 2}}$, then the maximum value of $xy$, for all $x \geq 0,\ y \geq 0$ is

Explanation

Reviewed by Dr. Abubacker

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