Answer:
The numbers are (40,-40).
Step-by-step explanation:
Given : Two numbers whose difference is 80 and whose product is a minimum.
To find : The numbers ?
Solution :
Let the largest number be x,
and smallest number be y.
According to question,
Two numbers whose difference is 80.
i.e. [tex]x-y=80[/tex] ......(1)
Product of two number is minimum.
i.e. [tex]xy=P[/tex] .....(2)
where P is minimum.
Now substitute x from (1) in (2),
[tex]x(x-80)=P[/tex]
[tex]x^2-80x=P[/tex]
Apply completing the square by adding 40 square both side,
[tex]x^2-80x+(40)^2=P+(40)^2[/tex]
[tex](x-40)^2=P+(40)^2[/tex]
The vertex of the equation occur at x=40.
So, the value of y is [tex]40-y=80[/tex]
[tex]y=-40[/tex]
Therefore, the numbers are (40,-40).
