Question

If ${\log }_{10}(x^3+y^3)-{\log }_{10}(x^2+y^2-xy)\le 2$ where x, y are positive real numbers Then the maximum value of $xy$ is

• A. $2500$
• B. $3000$
• C. $1200$
• D. $3500$

Answer 1

The correct option is A $2500$

${\log }_{10}(x^3+y^3)-{\log }_{10}(x^2+y^2-xy)\le 2⟹{\log }_{10}\frac{(x^3+y^3)}{(x^2+y^2-xy)}\le {\log }_{10}100⟹\frac{(x^3+y^3)}{(x^2+y^2-xy)}\le 100⟹\frac{(x+y)(x^2+y^2-xy)}{(x^2+y^2-xy)}\le 100⟹x+y\le 100Sincemaximumvalueofxyistobecalculated,wewillassumex+y=100Themaximumofxywhenx+y=100isobtainedonlywhenx=y=50.Hence,xy=2500.$