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2x + y = 12 (given). ====>
y = 12-2x
xy = x*(12-2x) = -2x^2 + 12x
This is an equation of the quadratic function (parabola). Since it has negative leading coefficient at x^2,
this parabola is turned downward and has a maximum.
For a general equation of a quadratic function q(x) = ax^2 + bx + c with a < 0,
it has a maximum at x = .
In your case a= -2, b= 12 and c= 0, so your parabola achieves its maximum at x= = 3.
And the value of the maximum is the value of this quadratic function at x= 3:
= -18 + 36 = 18.
Answer. The largest possible value of the product xy at given condition is 18.
Solved.
Plot y =
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On finding maximum/minimum of a quadratic function see my lessons in this site
- 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
- A rectangle with a given perimeter which has the maximal area is a square
- A farmer planning to fence a rectangular garden to enclose the maximal area
- A farmer planning to fence a rectangular area along the river to enclose the maximal area
- A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
- Using quadratic functions to solve problems on maximizing revenue/profit
- OVERVIEW of lessons on finding the maximum/minimum of a quadratic function
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.