If a+b=2007 then what is the maximum value of ab? (do not use caculus to solve)?

Question

do not use caculus to solve, and what values of a and b give the maximum product.

Answer 1

Logically the maximum product will be when a and b are exactly half of 2007.

a = b = 1003 ½
ab = 1,007,012.25

Let's let x = a
Then 2007 - x = b

Multiply these two together (ab):
x(2007 - x)
2007x - x²

So you have a graph:
y = -x² + 2007x

If you graph this you'll see the maximum point happens when x = 1003.5 and y = 1007012.25

Or the other thing you can do is complete the square, take half of the x-term (1003.5), square it (1007012.25) and add it and subtract it.
y = -x² + 2007x + 1007012.25 -1007012.25
y = -(x² + 2(-1003.5)x + (-1003.5)²) + 1007012.25

Now you have something you can factor to a perfect square:
y = -(x - 1003.5)² + 1007012.25

By observation, this sum will be maximum when you subtract the littlest amount possible. In fact, you want to subtract nothing. That will happen when:
(x - 1003.5) is zero, because you'll square it and subtract it and you'll take nothing away.

x - 1003.5 = 0
x = 1003.5

That's the long way, but you'll quickly realize that the quick, intuitive way is to always take half and you'll have the maximum product.

Answer 2

Without calculus.

Let
a = x + y
b = x - y

a + b = (x + y) + (x - y) = 2x = 2007
x = 2007/2 = 1003.5

ab = (x + y)(x - y) = x² - y²

Since squares are always positive or zero, the product of ab is greatest when y² is least. And that is when y = 0.

Therefore:

a = x + y = 1003.5 + 0 = 1003.5
b = x - y = 1003.5 - 0 = 1003.5

Maximum ab:

ab = 1003.5² = 1,007,012.25

Answer 3

ab
= (1/4) 4ab
=(1/4) [ (a + b)^2 - (a - b)^2 ]

Thus, ab is maximum when a - b = 0
and the maximum value is
(1/4)(a + b )^2
=(1/4)(2007)^2
= 1007012.25

Answer 4

Max value = (2007/2)^2 = 1007012.25

Answer 5

It will be (1003.5)^2. I know this because I have used calculus to solve similar problems; I know of no way to solve the problem without using calculus.

Answer 6

what the above people said

Answer 7

half of 2007

1003.5+1003.5= 2007

1003.5*1003.5 = 1007012.25

proof

1004*1003=1007012 ------ 0.25 less