Find the minimum value of the function f(x,y= 25x^2+3y^2+20xy+10x+4y+10 subject to the constraint 5x^2+4xy=1.
I solved fx=50x+20y+10
fy=6y+20x+4
gx=10x+4y
gy=4x
Then what?
I will solve it by Lagrange multipliers, although there may be other possible ways, depending on what you've done in school.
Let
f(x,y)=25x^2+3y^2+20xy+10x+4y+10 (function to be minimized)
c(x,y)=5x^2+4xy-1 (constraint = 0)
we will set
p(x,y)=f(x,y)+λ*c(x,y)
where λ is the Lagrange multiplier (lambda), to be determined.
We will calculate the partial derivatives px, py and equate to zero.
Together with the constraint c(x,y)=0, we have three equations in three unknowns (x,y,λ).
Solving the 3 non-linear equations will give x=1/sqrt(5), y=0, and λ=-sqrt(5)-5.
The minimum value of f(x,y) is 15+2√5.
Here are the details:
p(x,y)=(4xy+5x^2-1)λ+3y^2+20xy+4y+25x^2+10x+10
px(x,y)=(4y+10x)λ+20y+50x+10 = 0
py(x,y)=4xλ+6y+20x+4 = 0
c(x,y)=4xy+5x^2-1 = 0
Solve for x, y & λ from the three non-linear equations above to get
x=1/sqrt(5), y=0, λ=-sqrt(5)-5.
Would I then plug in x and y into the original equation to find the minimum value? When I do that I get approximately 19.472, but my webwork homework says that is incorrect.
You're right, the answer set
x=1/sqrt(5), y=0, λ=-sqrt(5)-5.
does not give the minimum.
I have taken the wrong set of answers which should have read:
x=-1/sqrt(5), y=0, λ=sqrt(5)-5
which gives f(x,y)=10.528 approx.
but unfortunately this one does not seem to be right either.
I'll look at it and get back to you.
In the mean time, you could repost and see if other teachers could give you an answer before I get back.
x=-1/sqrt(5), y=0, λ=sqrt(5)-5
which gives
f(x,y)=15-2√5
=10.528 approx.
should be the correct answer.
When I varied the values of x and y, I was able to get values of f(x,y) smaller than 15-2√5, which should not happen.
However, by varying the values of x and y, we no longer satisfy the constraints.
x=-1/sqrt(5), y=0 gives the minimum value when the constraints are satisfied.
That worked! Thank you very much. I spent a long time trying to solve that equation before I put it on jiskha. It was driving me crazy. I appreciated all your help!
You're very welcome!
Please do rework the problem and understand every step. It may help your exams!
If there is any doubt, post again!
Great job on finding the partial derivatives of the function f(x, y) and the constraint equation g(x, y). Now, to find the minimum value of the function f(x, y) subject to the constraint g(x, y), we can use the method of Lagrange multipliers.
1. Start by setting up the Lagrangian function L(x, y, λ) by combining the objective function f(x, y) with the constraint equation g(x, y):
L(x, y, λ) = f(x, y) - λ * g(x, y)
In this case, the Lagrangian function would be:
L(x, y, λ) = 25x^2 + 3y^2 + 20xy + 10x + 4y + 10 - λ * (5x^2 + 4xy - 1)
2. Take the partial derivatives of the Lagrangian function with respect to x, y, and λ, and set them equal to zero:
∂L/∂x = 0 -> 50x + 20y + 10 - λ(10x + 4y) = 0 (Equation 1)
∂L/∂y = 0 -> 6y + 20x + 4 - λ(4x) = 0 (Equation 2)
∂L/∂λ = 0 -> 5x^2 + 4xy - 1 = 0 (Equation 3)
3. Solve the above system of equations (Equations 1, 2, and 3) simultaneously. This system of equations is non-linear and might require solving with numerical methods like substitution or elimination. You can use methods like the Newton-Raphson method or numerical solvers in software like MATLAB or Python to find the solutions.
4. Once you find the solutions (x, y, λ), substitute them back into the original objective function f(x, y) to find the minimum value.
Note: Since the system of equations is non-linear, finding an exact solution may be difficult or time-consuming. Consider using numerical methods to approximate the solution.
I hope this helps! Let me know if you have any other questions.