Among all pairs of numbers whose difference is 88, find a pair whose product is as small as possible.
I bet one of the numbers is zero.
-44 and 44.
This can also be solved by doing:
x-y = 88
min x*y.
x = 88+y, so you get
minimize x^2+88x,
which can be done with calculus.
To find a pair of numbers whose difference is 88 and whose product is as small as possible, we can start by setting up an equation.
Let's assume the two numbers are x and y. According to the problem, the difference between these two numbers is 88, which gives us the equation:
x - y = 88
Now, to find the pair of numbers whose product is as small as possible, we need to minimize the product xy.
To do that, we can solve the equation for one variable in terms of the other and substitute it back into the product xy equation. Let's solve the equation for x in terms of y:
x = y + 88
Now, substitute this value of x into the product equation:
xy = (y + 88)y
Expanding the equation:
xy = y^2 + 88y
To minimize the product xy, we can find the minimum point of the quadratic equation by finding the vertex. The x-coordinate of the vertex of a quadratic equation of the form ax^2 + bx + c is given by x = -b/2a.
In our equation, a = 1, b = 88, and c = 0. Plugging these values into the formula, we get:
y = -(88) / 2(1)
y = -44
So, one of the numbers is -44 and the other number can be found by substituting the value of y into the equation we got when solving for x:
x = y + 88
x = -44 + 88
x = 44
Therefore, the pair of numbers whose difference is 88 and whose product is as small as possible is (-44, 44).