2
$\begingroup$

Let $f(x,y)=xy+ \frac{50}{x}+\frac{20}{y}$, Find the global minimum / maximum of the function for $x>0,y>0$

Clearly the function has no global maximum since $f$ is not bounded. I have found that the point $(5,2)$ is a local minimum of $f$. It seems pretty obvious that this point is a global minimum, but I'm struggling with a formal proof.

$\endgroup$
3
$\begingroup$

By AM-GM $$f(x,y)\geq3\sqrt[3]{xy\cdot\frac{50}{x}\cdot\frac{20}{y}}=30.$$ The equality occurs for $$xy=\frac{50}{x}=\frac{20}{y}=10,$$ id est, for $(x,y)=(5,2)$, which says that $30$ is a minimal value.

The maximum does not exist. Try $x\rightarrow0^+$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.