Hello, samsum!

I'll do part (c) . . .

Find all the relative maxima, minima and saddle points.

. .

Set the partial derivatives equal to zero and solve.

. .

Substitute (1) into (2): .

. . which has roots: .

Second Derivative Test: .

Then: .

At . . . saddle point at

At . . . minimum at

At . . . minimum at

2) Determine the dimensions of an open-top rectangular box having a volume of 32 ft³

requiring the least amount of material for its construction.

Let = length, = width, = height.

The volume is: .[1]

The surface area is: .[2]

Substitute [1] into [2]: .

. . and we have: . . . . which we will minimize.

Set the partial derivatives equal to zero and solve.

. .

Substitute (3) into (4): .

Substitute into (3): .

Substitute into (1): .

Therefore, the dimensions of the box are: . feet.