Consider two nonnegative numbers x and y such that
x + y = 12.
Find the values of x and y that optimize the following quantity.
xy
Maximize the quantity.
{x, y} =
Minimize the quantity. (Order your answers from smallest to largest x, then from smallest to largest y.)
{x, y}
=
{x, y}
=
To optimize the quantity xy, we can use the method of optimization by substitution.
Given that x + y = 12, we can solve for one variable in terms of the other and substitute it into the quantity xy.
Let's solve the equation x + y = 12 for y.
y = 12 - x
Substituting y = 12 - x into the quantity xy:
xy = x(12 - x)
Now, we have a single variable expression. We can find the critical points of the function xy by taking the derivative and setting it equal to zero.
d(xy)/dx = 12 - 2x
Setting the derivative equal to zero:
12 - 2x = 0
Solving for x:
2x = 12
x = 6
Now that we have the value of x, we can substitute it back into the equation y = 12 - x to find the value of y.
y = 12 - 6
y = 6
Thus, the critical point for maximizing the quantity xy is at {x, y} = {6, 6}.
To minimize the quantity xy, we need to consider the end points of the feasible region. The non-negative constraint on x and y implies that the lower bound for both variables is 0.
So, the smallest possible values for x and y are 0.
Thus, the minimum value of the quantity xy is 0, and the values of x and y that minimize the quantity are {x, y} = {0, 0}.
To find the values of x and y that optimize the quantity xy, we need to maximize and minimize the value of xy while satisfying the given condition x + y = 12.
To maximize the quantity xy, we can use the concept of arithmetic mean-geometric mean inequality (AM-GM inequality). According to the AM-GM inequality, the maximum value of the product xy occurs when x and y are equal to each other.
Since x + y = 12, we can rewrite it as 2x = 12, which implies x = 6. Therefore, y = 12 - 6 = 6. So, to maximize the quantity xy, the values of x and y should be equal to each other, which is {x, y} = {6, 6}.
To minimize the quantity xy, we can use the same idea. According to the AM-GM inequality, the minimum value of the product xy occurs when x and y are as unequal as possible. In this case, the smallest possible value for one variable is 0.
If we set x = 0, then from the equation x + y = 12, we can solve for y as y = 12 - x = 12 - 0 = 12. So, one possible combination to minimize the quantity xy is {x, y} = {0, 12}.
If we set y = 0, then from the equation x + y = 12, we can solve for x as x = 12 - y = 12 - 0 = 12. So, another possible combination to minimize the quantity xy is {x, y} = {12, 0}.
Therefore, the solutions for minimizing the quantity xy are {x, y} = {0, 12} and {12, 0}.
xy = x(12-x)
that is a parabola with vertex at x=6
This is just an illustration that the rectangle with the largest area for a given perimeter is a square.
clearly the minimum is achieved when the rectangle is just a line of length 12 and area zero.