Answer
$x=55$
$y=55$
$P=3025$
Work Step by Step
Quadric function in standard form:
$y=a(x-h)^2+k$, in which $(h,k)$ is the vertex. And, the maximum (or the minimum) occurs at the vertex.
Two positive real numbers: $x$ and $y$
$x+y=110$
$y=110-x$
Product:
$P=xy$
$P=x(110-x)=110x-x^2=-x^2+110x$
$P=-(x^2-110x)$
$P=-[(x^2+2(55)x+55^2)-55^2]$
$P=-(x-55)^2+3025~~$ (Notice that: $a=-1$, so the parabola opens downward)
So the vertex $(55,3025)$ is the maximum.
$y=110-55=55$
