Question

Find the numbers whose product is 100 and whose sum is minimum.

Answer 1

Let the numbers be $x,y$

$xy=100,y=\frac{100}{x}$

$f(x,y)=x+y$

$f\left(x\right)=x+\frac{100}{x}$

Minimum of $f\left(x\right)$ is obtained at $f^{\prime }\left(x\right)=0$

$\Rightarrow f^{\prime }\left(x\right)=1-\frac{100}{x^2}=0$

$x=\pm 10$

$\therefore x=10,y=10$