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

For more information, refer to the link given below:

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3201

step-by-step explanation:

use an equation of ratios to determine the amount of gas needed to fill the tank.

    0.2             0.8

  =    

288.75             x

                                                              231 gal

so, 0.2x = (0.8)(288.75 gal), and x = = 1155 gal.

                                                                0.2

it would take 1155 gal. to fill the tank.

the cost would be

1155 gal         $3

* = $3465   (answer)

      1           1 gal

hope this you