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if x + y = 50, what is the maximum value of the product of x and y that is possible?
Given: $x + y =50$
To find: Maximum value of $x \times y$
$x + y =50$
$y =50 - x$
Substitute in $x \times y$
= $x \times (50-x) $
= $50x -x^2 $
= $50x -x^2 $
If $a < 0$, quadratic expression has greatest value at $x = – \frac{b}{2a}$
Our quadratic expression has $a = -1$ and $b = 50$. So greatest value is at $x = – \frac{50}{-2}$
Greatest value is at $x =25$
Substitute $x =25$
We get
=$50 \times 25 -25^2 $
= $25(50-25)$
= $25(25)$
= 625
So maximum value is 625
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