SOLUTION: The sum of one number and two times a second number is 24. What numbers should be selected so that their product is as large as possible?

SOLUTION: The sum of one number and two times a second number is 24. What numbers should be selected so that their product is as large as possible?

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Question 797678: The sum of one number and two times a second number is 24. What numbers should be selected so that their product is as large as possible?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

We are interested in maximizing the product function:

Substituting:

Which is the function we need to maximize.

Algebra Way

Put the quadratic function into standard form:

So we have

,

, and

Since this is a quadratic function with a negative lead coefficient, the graph is a parabola that opens downward. Since it opens downward, the vertex is a maximum point. The

-coordinate of the vertex is given by:

In this case:

Hence the value of

for the maximum product is 6, so

Calculus way

Take the first derivative of the product function:

Set the first derivative equal to zero and solve:

Hence the function has a critical point at

Take the second derivative

Since the second derivative is less than zero for all

in the domain of

, the critical point is a maximum.

Hence, the product is maximum when

and

John

Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it

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