Quadratic Functions

Question

Find algebraically two numbers whose sum is 50 and whose product is a maximum.

Answer 1

Let x and y be two numbers.

x+y=50

We have to find the maximum of the product (xy). Let consider a function f(x)= xy. Let solve the equation above for y:

y=50-x. Let's substitute y in the equation of function f(x):

f(x)= x(50-x)=50x-x^2.

We have quadratic equation. Because, the coefficient in front of x^2 is negative, the parabola will have the maximum at the vertex. So, we need to find the coordinates of vertex.The formula for finding the x ccordinate of the vertex is -b/2a, where a and b are coefficients of the equation of parabola: ax^2+bx+c=f(x).

-b/2a=-50/(2(-1))=25.

So, at x=25 the function f(x) will have the maximum. Solution : x=25, y=25.