Question

The sum of two integers is 16. find the values of these two integers so that their product is a maximum, and find the maximum value of the product.

Answer 1

$x,y-\text{the integers}$

$\left\{\begin{array}{ccc}x+y=16\\xy=max\end{array}\right\\\\x+y=16\to y=16-x\\\\\text{substitute to the product}\\\\x(16-x)=-x^2+16x\\\\\text{we have the quadratic equation-quadratic function}\\\\f(x)=-x^2+16x\\\\\text{the maximum is in the vertex}\\\\\text{the formula of the vertex form}\\\\f(x)=a(x-h)^2+k\\\\(h;\ k)\ \text{the coordinates of the vertex}$
$f(x)=-x^2+16x=-(x^2-16x)=-(x^2-2\cdot x\cdot8)\\\\=-(\underbrace{x^2-2\cdot x\cdot8+8^2}_{(a-b)^2=a^2-2ab+b^2}-8^2)=-[(x-8)^2-64]\\\\=-(x-8)^2+64\\\\h=8\ \text{and the maximum is equal 64} \\\\\text{The integers}\\x=8;\ y=16-8=8\\\\\text{The maximum is equal}\ 64$