5 Answers

If two numbers add up to a certain amount then the biggest product is always when the two numbers are the same.
In this case …
Half168 is 84.
So the numbers are 84 and 84
because the product is MINUS 7056 !

Rewrite using math symbols.
Assuming largest number is x
x – y = 168 ==> y = x – 168
xy = P where P is a minimum
x(x168) = P. This is a parabola.
We can find vertex (minimum) by putting in vertex format.
Use completing square technique
x2 – 168x = P
x^2 – 168x + 7056 = P + 7056 (add 7056 to complete square)
(x – 84)^2 = P + 7056.
Vertex occurs where x = 84. y = 84 168 = 84.
Answer 84, 84.

But if you are not in calculus:
x = one number
y = the other (higher)
y – x = 168
y = 168 + x
product = xy = x(168 + x) = 168x + x^2 = 1x^2 + 168x
If y = ax^2 + bx + c and a is positive,
a minimum occurs when x = b / (2a)
so x = (168) / (2 • 1) = 168/ 2 = 84
and y = 168 + x = 168 + (84) = 84

y = x168
p = product = xy = x^2168x
dP/dx = 2x 168 = 0
x = 84
y = 84
Numbers are 84 and 84

x – y = 168
P = x * y
P = x * y
P = (168 + y) * y
P = 168 * y + y^2
dP/dy = 168 + 2y
dP/dy = 0
0 = 168 + 2y
0 = 84 + y
y = 84
x – y = 168
x – (84) = 168
x + 84 = 168
x = 84
84 and 84