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):
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
