Respuesta :
Lagrange multipliers:
[tex]L(x,y,z,\lambda)=xy^2z^2+\lambda(x+y+z-5)[/tex]
[tex]L_x=y^2z^2+\lambda=0[/tex]
[tex]L_y=2xyz^2+\lambda=0[/tex]
[tex]L_z=2xy^2z+\lambda=0[/tex]
[tex]L_\lambda=x+y+z-5=0[/tex]
[tex]\lambda=-y^2z^2=-2xyz^2=-2xy^2z[/tex]
[tex]-y^2z^2=-2xyz^2\implies y=2x[/tex] (if [tex]y,z\neq0[/tex])
[tex]-y^2z^2=-2xy^2z\implies z=2x[/tex] (if [tex]y,z\neq0[/tex])
[tex]-2xyz^2=-2xy^2z\implies z=y[/tex] (if [tex]x,y,z\neq0[/tex])
In the first octant, we assume [tex]x,y,z>0[/tex], so we can ignore the caveats above. Now,
[tex]x+y+z=5\iff x+2x+2x=5x=5\implies x=1\implies y=z=2[/tex]
so that the only critical point in the region of interest is (1, 2, 2), for which we get a maximum value of [tex]f(1,2,2)=16[/tex].
We also need to check the boundary of the region, i.e. the intersection of [tex]x+y+z=5[/tex] with the three coordinate axes. But in each case, we would end up setting at least one of the variables to 0, which would force [tex]f(x,y,z)=0[/tex], so the point we found is the only extremum.
[tex]L(x,y,z,\lambda)=xy^2z^2+\lambda(x+y+z-5)[/tex]
[tex]L_x=y^2z^2+\lambda=0[/tex]
[tex]L_y=2xyz^2+\lambda=0[/tex]
[tex]L_z=2xy^2z+\lambda=0[/tex]
[tex]L_\lambda=x+y+z-5=0[/tex]
[tex]\lambda=-y^2z^2=-2xyz^2=-2xy^2z[/tex]
[tex]-y^2z^2=-2xyz^2\implies y=2x[/tex] (if [tex]y,z\neq0[/tex])
[tex]-y^2z^2=-2xy^2z\implies z=2x[/tex] (if [tex]y,z\neq0[/tex])
[tex]-2xyz^2=-2xy^2z\implies z=y[/tex] (if [tex]x,y,z\neq0[/tex])
In the first octant, we assume [tex]x,y,z>0[/tex], so we can ignore the caveats above. Now,
[tex]x+y+z=5\iff x+2x+2x=5x=5\implies x=1\implies y=z=2[/tex]
so that the only critical point in the region of interest is (1, 2, 2), for which we get a maximum value of [tex]f(1,2,2)=16[/tex].
We also need to check the boundary of the region, i.e. the intersection of [tex]x+y+z=5[/tex] with the three coordinate axes. But in each case, we would end up setting at least one of the variables to 0, which would force [tex]f(x,y,z)=0[/tex], so the point we found is the only extremum.
The points on the portion of the plane x+y+z=5 in the first octant at which [tex]\rm f(x,y,z) = xy^2z^2[/tex] has a maximum value are (1,2,2) and this can be determined by using the Lagrange Multipliers.
Given :
- The portion of the plane x+y+z=5 in the first octant.
- [tex]\rm f(x,y,z) = xy^2z^2[/tex]
According to the Lagrange Multipliers:
[tex]\rm L(x,y,z,\lambda ) = xy^2z^2+\lambda(x+y+z-5)[/tex]
Now, the values become:
[tex]\rm L_x = y^2z^2+\lambda[/tex]
[tex]\rm L_y = 2xyz^2+\lambda[/tex]
[tex]\rm L_z = 2xy^2z+\lambda[/tex]
[tex]\rm L_{\lambda} = x+y+z-5[/tex]
[tex]\rm \lambda = -y^2z^2=-2xyz^2=-2xy^2z[/tex]
From the above equation, it can be concluded that:
y = 2x (if y ,z [tex]\neq[/tex] 0)
z = 2x (if y ,z [tex]\neq[/tex] 0)
z = y (if x, y, z [tex]\neq[/tex] 0)
Now, substitute the values of known terms in (x + y + z = 5).
2x + 2x + x = 5
5x = 5
x = 1
So, the value of y = z = 2.
So, the maximum value of the function is:
f(1,2,2) = 16
For more information, refer to the link given below:
https://brainly.com/question/4609414
