If y = 18 - 3x, what is the maximum value of the product 2xy?
A. -3
B. 0
C. 3
D. 6
E. 54
I found that the value of the maximum occurs at x = 3, but that the value itself is 54. Is E the answer?
yes
To find the maximum value of the product 2xy, we need to find the value of x that maximizes the expression.
The given equation is y = 18 - 3x.
We can substitute this equation into the expression for 2xy:
2xy = 2x(18 - 3x).
To find the maximum value, we need to optimize this expression. Since it is a quadratic equation, we can use calculus to find the maximum.
Step 1: Take the derivative of the expression with respect to x:
d/dx (2x(18 - 3x)).
Using the product rule, we get:
2(18 - 3x) + 2x(-3).
Simplifying, we get:
36 - 6x - 6x.
Step 2: Set the derivative equal to zero and solve for x:
36 - 12x = 0.
Rearranging the equation, we find:
12x = 36.
Solving for x, we get:
x = 3.
Step 3: Substitute the value of x into the expression for 2xy to find the maximum value:
2xy = 2(3)(18 - 3(3)).
Simplifying, we get:
2xy = 2(3)(18 - 9).
2xy = 2(3)(9).
2xy = 6(9).
2xy = 54.
Therefore, the maximum value of the product 2xy is 54.
Therefore, the correct answer is E.
Yes, you are correct. The value of the maximum product occurs at x = 3, and the maximum value of the product 2xy is indeed 54. Therefore, the correct answer is option E.
To find the maximum value, you first need to substitute the expression for y in terms of x into the expression for the product 2xy:
2xy = 2x(18 - 3x) = 36x - 6x^2
The maximum value occurs at the vertex of the quadratic function, which corresponds to the x-value that gives the maximum product. To find this value, you can use the formula x = -b/2a, where the quadratic equation is in the form ax^2 + bx + c. In this case, a = -6, b = 36, and c = 0:
x = -b/2a = -36/(2*(-6)) = -36/(-12) = 3
So, the maximum value occurs when x = 3. Substituting this value back into the expression for 2xy gives:
2xy = 2(3)(18 - 3(3)) = 2(3)(18 - 9) = 2(3)(9) = 54
Thus, the maximum value of the product 2xy is indeed 54, and the correct answer is option E.