Solve f(x,y,z): Min & Max Values Subject to Constraint x^2+2y^2+6z^2=81

In summary, the function has a minimum value at $x=\frac{2x}{9}$ and a maximum value at $x=\frac{3x}{12}$.
  • #1
ajkess1994
9
0
Afternoon, I have been working on this problem for awhile now but have been stuck on a certain point, and once I set everything equal to each other I end up with the same thing for example: x=x, y=y, & z=z

Find the minimum and maximum values of the function f(x,y,z)=3x+2y+4z subject to the constraint x^(2)+2y^(2)+6z^(2)=81.

fmax= ?
fmin= ?

This is what I have,

fx=3, fy=2, fz=4, gx=2x, gy=4y, gz=12z

3=2x(lamda); (lamda)= 3/2x

2=4y(lamda); (lamda)= 2/4y

4=12y(lamda); (lamda)= 4/12z

(3/2x)=(2/4y)=(4/12z)
 
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  • #2
ajkess1994 said:
Afternoon, I have been working on this problem for awhile now but have been stuck on a certain point, and once I set everything equal to each other I end up with the same thing for example: x=x, y=y, & z=z

Find the minimum and maximum values of the function f(x,y,z)=3x+2y+4z subject to the constraint x^(2)+2y^(2)+6z^(2)=81.

fmax= ?
fmin= ?

This is what I have,

fx=3, fy=2, fz=4, gx=2x, gy=4y, gz=12z

3=2x(lamda); (lamda)= 3/2x

2=4y(lamda); (lamda)= 2/4y

4=12y(lamda); (lamda)= 4/12z

(3/2x)=(2/4y)=(4/12z)

Hi ajkess1994 and welcome to MHB! ;)

Express $y$ in $x$, and also $z$ in $x$, and substitute in $x^2+2y^2+6z^2=81$?
 
  • #3
ajkess1994 said:
Afternoon, I have been working on this problem for awhile now but have been stuck on a certain point, and once I set everything equal to each other I end up with the same thing for example: x=x, y=y, & z=z

Find the minimum and maximum values of the function f(x,y,z)=3x+2y+4z subject to the constraint x^(2)+2y^(2)+6z^(2)=81.

fmax= ?
fmin= ?

This is what I have,

fx=3, fy=2, fz=4, gx=2x, gy=4y, gz=12z

3=2x(lamda); (lamda)= 3/2x

2=4y(lamda); (lamda)= 2/4y

4=12y(lamda); (lamda)= 4/12z

(3/2x)=(2/4y)=(4/12z)

I would first look at:

\(\displaystyle \frac{3}{2x}=\frac{2}{4y}\implies y=\frac{x}{3}\)

And then:

\(\displaystyle \frac{3}{2x}=\frac{4}{12z}\implies z=\frac{2x}{9}\)

Now, substitute into your constraint, and solve for \(x\)...what do you get?
 
  • #4
Thank you MarkFL the process carried out and worked.
 

FAQ: Solve f(x,y,z): Min & Max Values Subject to Constraint x^2+2y^2+6z^2=81

1. What is the objective function in this problem?

The objective function in this problem is f(x,y,z), which represents the function that we are trying to optimize (either minimize or maximize).

2. What is the constraint in this problem?

The constraint in this problem is x^2+2y^2+6z^2=81, which is an equation that limits the values of x, y, and z that can be used in the objective function.

3. How many variables are there in this problem?

There are three variables in this problem: x, y, and z.

4. How do you find the minimum and maximum values of the objective function?

To find the minimum and maximum values of the objective function, we first need to find the critical points by taking the partial derivatives of the objective function with respect to each variable and setting them equal to 0. Then, we use the second derivative test to determine whether each critical point is a minimum or maximum. Finally, we evaluate the objective function at these points to find the minimum and maximum values.

5. Can this problem be solved using calculus?

Yes, this problem can be solved using calculus. The method described in the previous answer is known as the method of Lagrange multipliers and is commonly used to optimize a function subject to a constraint.

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