Find all points on the portion of the plane x+y+z=5 in the first octant at which f(x, y, z) = xy2z2 has a maximum value.

The points on the portion of the plane x+y+z=5 in the first octant at which 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.

According to the Lagrange Multipliers:

Now, the values become:

From the above equation, it can be concluded that:

y = 2x (if y ,z 0)

z = 2x (if y ,z 0)

z = y  (if x, y, z 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:

link

Lagrange multipliers:










(if )

(if )

(if )

In the first octant, we assume , so we can ignore the caveats above. Now,



so that the only critical point in the region of interest is (1, 2, 2), for which we get a maximum value of .

We also need to check the boundary of the region, i.e. the intersection of 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 , so the point we found is the only extremum.

Arhombus is a quadrilateral.

I’m sure you could google it. my profile picture might