Max min values of a particular function

Question

If $xy=10$ then what is the minimum value of $12x^2+13y^2$?

What should the general approach be in these type of problems?

Answer 1

When you search an extremum relatively to a condition you can also use Lagrange multipliers.

Search minimum of $f(x,y)=12x^2+13y^2$ given condition $g(x,y)=xy-10=0$

Then $\lambda\Delta g=\lambda(y,x)=\Delta f=(24x,26y)$

And you'll find $\lambda^2xy=24\times 26\times xy$ or $\lambda=4\sqrt{39}$

From there you can find that the minimum is reached for $12x^2=13y^2$ with value $\lambda xy=40\sqrt{39}$

Answer 2

With $$y=\frac{10}{x}$$ you will get $$f(x)=12x^2+13\left(\frac{10}{x}\right)^2$$

Answer 3

Rescaling:

$u:=\sqrt{12}x$; $v:=\sqrt{13}y.$

We look for the maximum of

$u^2 +v^2$ with constraint

$uv =(10)\sqrt{(12)(13)}$:

$u^2+v^2 \ge 2|uv| = (20)2\sqrt{39}.$

Used:

$u^2+ v^2 \ge 2|uv|.$

Can you prove the above inequality?