Can someone please walk me through the steps, I am just not sure what to do next. thanks
Solve the optimization problem.
Minimize F = x^2 + y^2 subject to xy^2 = 16
I took the derivative F = 2x + 2Y but I don't know where to go from here.
from xy^2 = 16
y^2 = 16/x
now sub that into
F = x^2 + y^2
= x^2 + 16/x
dF/dx = 2x - 16/x^2
= 0 for max/min
2x - 16/x^2 = 0
2x = 16/x^2
2x^3 = 16
x^3 = 8
x = cuberoot(8) = 2
if x = 2 , y^2 = 16/2 = 8 ---> y = ±√8 or ±2√2
then F = x^2 + y^2
= 4 + 8 = 12
by 2nd derivative test
F '' = 2 + 32/x^3 which is > 0 for x = 2
so when x = 2, F is a minimum of 12
To solve the optimization problem, we need to find the critical points by setting the partial derivatives equal to zero. In your case, you already found the partial derivatives:
∂F/∂x = 2x
∂F/∂y = 2y
The next step is to set these partial derivatives equal to zero and solve for x and y:
2x = 0 ----> x = 0
2y = 0 ----> y = 0
Now, we need to check if these critical points satisfy the constraint xy^2 = 16. Plugging in the values:
x * y^2 = 16
0 * y^2 = 16 ---> 0 = 16
Since 0 does not equal 16, (0, 0) is not a valid critical point.
To proceed further, we can use the constraint xy^2 = 16 to express one variable in terms of the other. We can solve for x:
xy^2 = 16
x = 16/y^2
Now, substitute this expression for x in the objective function F = x^2 + y^2:
F = (16/y^2)^2 + y^2
F = 256/y^4 + y^2
To minimize F, we can take the derivative of F with respect to y and set it equal to zero. So, take the derivative:
dF/dy = -1024/y^5 + 2y = 0
Now, we solve this equation for y. Let's re-arrange the equation:
1024/y^5 = 2y
Multiply both sides by y^5:
1024 = 2y^6
Divide both sides by 2:
512 = y^6
Take the sixth root of both sides:
y = (512)^(1/6)
Now, substitute this y value back into the constraint xy^2 = 16 to find the corresponding x value:
x * (512)^(1/6)^2 = 16
x * (512)^(1/3) = 16
Solve for x:
x = 16 / (512)^(1/3)
Now, you have the values of x and y that minimize the objective function F = x^2 + y^2 subject to the constraint xy^2 = 16.