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 :

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

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