Find two numbers whose difference is 132 and whose product is a minimum.
smaller number --- x
larger number ----- x + 132
product=P=x(x+132)
= x^2 + 132x
dP/dx = 2x + 132
=0 for a min of P
2x = -132
x = -66
the smaller is -66
the larger -66+132 or +66
the two numbers are -66 and +66
its very smart way for a explanation but doesn't really make sense
Why not make it a family affair and call them "Mr. Subtract" and "Mrs. Multiply." They have a lovely little baby called "Product." It's safe to say, they don't need any more additions to their numerically inclined family!
To find two numbers whose difference is 132 and whose product is a minimum, we can first express the numbers in terms of a variable and set up an equation based on the given information.
Let's assume one of the numbers is x, then the other number would be x + 132 (since their difference is 132).
The product of these two numbers is given by multiplying them together: x(x + 132) = x^2 + 132x.
To find the minimum product, we need to find the vertex of the quadratic equation x^2 + 132x. The x-coordinate of the vertex gives us the number with which we can minimize the product.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a = 1 (the coefficient of x^2) and b = 132 (the coefficient of x).
Substituting the values for a and b, we have x = -132 / (2 * 1) = -132 / 2 = -66.
So, one of the numbers is -66, and the other number is -66 + 132 = 66.
Therefore, the two numbers whose difference is 132 and whose product is a minimum are -66 and 66.