The question can be answered quickly, by considering that we have only two numbers that are multiplied with each other. So if the numbers are represented by a and b, with the constraint that a+b=24 and

`a*b=max`

Evidently these two numbers that are multiplied are maximum when they are equal, because in this case we have the square of the number. That is, for a=b, 2a = 24 or a = 12 and a^2=144 which is maximum.

We can use calculus to come to the same conclusion.

a+b=24 means b=24-a

So ab = a*(24-a) = 24*a-a^ 2

This is a function of a, f(a) = 24*a - a^2

The first derivative in relation to a is

`(df)/(da)=24-2a`

The extremum of this function is calculated for this first derivative being equal to zero. Then a = 12 as before. To check that this is maximum point, evaluate the second derivative:

`(d^2f)/(da^2)=-2`

Since this is negative, it representa a extremum that is maximum.

Find the maximum product of two numbers if their sum is 24.

Without calculus:

Let x be one of the numbers; the other will be 24-x. Let y be the product:

`y=x(24-x) `

`y=-x^2+24x `

The graph of this function is a parabola opening down. The maximum will occur at the vertex.

The vertex lies on the axis of symmetry: if `y=ax^2+bx+c ` then the axis of symmetry is the line `x=(-b)/(2a) ` .

For our problem, the axis of symmetry is` ` `x=(-24)/(2(-1)) ` or x=12.

**The maximum occurs when x=12; the maximum is** `y=12(24-12)=144 ` .

(Note that this occurs when the two numbers are x=12 and (24-x)=12.

Let x and y represent the two numbers that add up to 24.

x + y = 24

=> y = 24 - x

The product of the two numbers is P = x*y = x*(24 - x) = 24x - x^2.

To maximize the product P, solve P' = 0 for x.

P' = 24 - 2x

24 - 2x = 0

=> x = 12

Also P'' = -2 which is negative for x = 12

The product when x = 12 is 12*12 = 144

**The maximum value of the product of two numbers that add up to 24 is 144.**