# Find two positive real numbers whose product is a maximum.

Find two positive real numbers whose product is a maximum. The sum is 110.
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Witionsion

Let x and y be the two positive real numbers. Write an equation to show that the sum of the two positive real numbers is 110.
$x+y=110$
Solve for the value of y.
$x+y=110$
$x+y-x=110-x$
$y=110-x$
Write an equation to show that the product of the two real numbers is the maximum. Let f(x) be the maximum. Substitute the values of x and y. Write as a quadratic function, $f\left(x\right)=a{x}^{2}+bx+c$
$f\left(x\right)=xy$
$f\left(x\right)=x\left(110-x\right)$
$f\left(x\right)=110x-{x}^{2}$
$f\left(x\right)=-{x}^{2}+110x$
$a=-1$ which is $a<0$, the maximum at $x=-\frac{b}{2a}$. Detemine the value of x. Let a=-1 b=110
$x=-\frac{b}{2a}$
$x=-\frac{110}{2\left(-1\right)}$
$x=-\frac{110}{-2}$
$x=55$
Substitute the value of x in $y=110-x$
$y=110-x$
$y=110-55$
$y=55$
The two positive real numbers are 55 and 55