Find two positive numbers whose sum is 15 such that the product of the first and the square of the second is maximal.
I came up with this so far:
x + y = 15
xy^2 is the maximum
derivative of xy^2= 2xyy' + y^2
Now how do I solve this ^ after I set it to zero? I am stuck on that. Thank you so much
Thank you! I solved it out, and I got x=5 and y= 10 with a product of 500. Is this correct
You need to substitute
y = 15-x
x(15-x)^2
x(225 -30x+x^2)
225x -30x^2 + x^3
Now you can take the derivative and set it equal to zero.
I agree. You are welcome.
To find the two positive numbers x and y that satisfy the given conditions, we can use the method of optimization.
1. Start with the equation x + y = 15, which represents the sum constraint.
2. We need to maximize the expression xy^2.
3. To find the maximum, let's consider setting up an optimization problem using the method of Lagrange multipliers. By introducing a Lagrange multiplier λ, we can consider the function L(x, y, λ) = xy^2 + λ(x + y - 15).
4. Now, let's take the partial derivatives of L with respect to x, y, and λ:
∂L/∂x = y^2 + λ = 0 ------ (1)
∂L/∂y = 2xy + λ = 0 ------ (2)
∂L/∂λ = x + y - 15 = 0 ------ (3)
5. Solve the above system of equations (1), (2), and (3) simultaneously.
From equation (1), we have y^2 + λ = 0.
From equation (2), we have 2xy + λ = 0.
Solving both equations, we get y^2 = -λ and 2xy = -λ.
Multiplying both sides of 2xy = -λ by y, we have 2xy^2 = -yλ.
Now, substitute y^2 = -λ into the equation above: 2xy^2 = 2x(-λ) = -yλ.
6. Rearrange the equation to get: yλ = -2xy^2.
7. From equation (3), we have x + y - 15 = 0.
8. Now, substitute yλ = -2xy^2 and x + y = 15 into the equation above:
x + y = 15 -----> y = 15 - x
Substituting into yλ = -2xy^2:
(15 - x)λ = -2x(15 - x)^2.
9. Simplify the equation further:
15λ - λx = -2x(15 - x)^2.
10. To solve this equation for x, differentiate both sides with respect to x:
15λ - λ = -2(15 - x)^2 + 4x(15 - x).
11. Simplify the equation:
15λ - λ = -2(15 - x)^2 + 4x(15 - x).
15λ - λ = -2(225 - 30x + x^2) + 60x - 4x^2.
15λ - λ = -450 + 60x - 2x^2 + 60x - 4x^2.
15λ - λ = -6x^2 + 120x - 450.
12. Set the equation equal to zero to solve for x:
15λ - λ = -6x^2 + 120x - 450 = 0.
13. Solve the quadratic equation for x to find the value(s) of x that satisfy the equation.
14. Once you have the value(s) of x, substitute it back into y = 15 - x to find the corresponding values of y.
15. Finally, check which pair (x, y) satisfies the conditions of being positive numbers and whose sum is 15.
Note: Solving the quadratic equation will give you the possible values of x. You will need to assess whether they are positive and calculate the corresponding values of y to determine which pair satisfies all the conditions.