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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.

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.

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!