SOLUTION: The sum of two numbers is 29. find the maximum possible product of the two numbers.

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Question 1179760: The sum of two numbers is 29. find the maximum possible product of the two numbers.
Found 2 solutions by mananth, ikleyn:
Answer by mananth(16946)   (Show Source):
You can put this solution on YOUR website!
Let one number x and the other one y.

x + y = 29

The product of the two numbers will be x * y = xy

solve x + y = 29

y = 29 - x

x * y = x(29 - x) = 29x - x²

let f(x)= 29x-x^2,
f'(x) = 29 - 2x

The maximum value of this function will occur at a critical number. A critical number occurs where f'(x) = 0 or is undefined.

f'(x) is defined for all real x.

We only need to determine where f'(x) = 0

29 - 2x = 0

29 = 2x

x = 14.5

The two numbers are 14.5 ,14.5 The maximum product is 210.5.

Answer by ikleyn(52798)   (Show Source):
You can put this solution on YOUR website!
.

If x is one number, then the other number is 29-x. Their product is x*(29-x) = 29x - x^2. (1) This function is a parabola placed downward. 29x - x^2 = . (2) The parabola has the vertex at x = 14.5. So, the two numbers are equal to 14.5. Their product is 14.5*14.5 = 210.25.

Solved and explained.

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On finding the maximum/minimum of a quadratic function see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola

Consider these lessons as your textbook, handbook, tutorials and (free of charge) home teacher.
Learn the subject from there once and for all.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.