What is the maximum product of two numbers that add up to give 24.

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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.

embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

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.

atsuzuki's profile pic

atsuzuki | (Level 1) Adjunct Educator

Posted on

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.

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