We want to find two whole numbers whose difference is 68 and whose product is minimum.
The two numbers are 34 and -34, and the product is:
P = -1,156
Let's say that our two numbers are A and B, such that A is the smaller one.
We know that the difference must be 68, then we have:
B - A = 68.
And the product between these two numbers can be written as:
P = B*A
If we isolate one of the variables in the first equation, we get:
B = 68 + A
Now we can replace that in the product to get;
P = (68 + A)*A = 68*A + A^2
We want to minimize this, notice that this is a quadratic polynomial of positive leading coefficient, meaning that the minimum is at the vertex.
And for the general quadratic polynomial:
y = a*x^2 + b*x + c
The vertex is at:
x = -b/2a
Then in this case the vertex is at:
A = -68/(2*1) = -34
The value of B is given by:
B = 68 + (-34) = 34
The product is:
P = 34*(-34) = -1,156
If you want to learn more, you can read:
https://brainly.com/question/3496582
