# X+y=50 which x and y are two positive numbers (a)for which numbers x is the product of the two numbers an increasing function of x (b)what is the maximum value of their product?

Mar 20, 2018

We can see that $y = 50 - x$. Let $p \left(x\right)$ be the product.

$p \left(x\right) = x y = x \left(50 - x\right) = 50 x - {x}^{2}$

Now we take the derivative.

$p ' \left(x\right) = 50 - 2 x$

Take the derivative at $0$.

$0 = 50 - 2 x$

$2 x = 50$

$x = 25$

Which means that the maximum product will happen when $x = y = 25$. This is a downward opening parabola which means anywhere left of the vertex is increasing, so the range is $0 < x < 25$.

Hopefully this helps!

Mar 20, 2018

See below.

#### Explanation:

Here the extremum point problem is equivalent to this one.

Given the function

$f \left(x , y\right) = x y - C = 0$

determine $C$ such that $f \left(x , y\right)$ and $x + y = 50$ are tangent.

Choosing $x = \frac{C}{y}$ and substituting into $x + y = 50$ we have

$\frac{C}{y} + y = 50$ or

${y}^{2} - 50 y + C = 0 \Rightarrow y = \frac{1}{2} \left(50 \pm \sqrt{{50}^{2} - 4 C}\right)$

but tangency implies one solution hence ${50}^{2} - 4 C = 0$

and finally

$C = {25}^{2} , y = 25 \Rightarrow x = 25$

b) ${25}^{2}$