Question 1180887: Find two numbers differing by 48 whose product is as small as possible. Found 2 solutions by ikleyn, MathLover1:Answer by ikleyn(52903) (Show Source):
If x is the smaller number, the the greater number is (x+48), according to the condition,
and the product of the two numbers is x*(x+48) = x^2 + 48.
They want you find the MINIMUM of this quadratic function.
The minimum is the vertex of the parabola, and on x-axis, this minimum is located
exactly half way between the zeroes of this quadratic function.
The zeroes are -48 and 0, so = = -24.
The minimal value of the function is y = (-24)*(-24+48) = (-24)*24 = -576.
ANSWER. The minimum value of the producr of such two numbers is -576.
It is achieved at the pair of the numbers (-24,24).
Visual check
Plot y = x*(x+48) (red line), y = -576 (green line)
You can put this solution on YOUR website!
let two numbers be and
if they differing by we have
..........eq.1
Their product is
.......eq.2, substitute
Since ,
minimize the function by calculating ' '
Setting it to , we get
=>
then ..........eq.1
so, your numbers are and
