Area D with six surfaces is stated as follows

Question

D=2xy+2yz+2zx

Volume is expressed as follow= xyz =1000

i) Find the area equation of area, D in x and y only.

ii) Find the critical point for the equation obtained in question.

iii) Determine whether it has a maximum or minimum area area.

Answer 1

i) z = 1000/(xy), so D = 2xy + 2000/x + 2000/y

ii) ∂D/∂x = 2y - 2000/x² = 0

∂D/∂y =2x - 2000/y² = 0

yx² = 1000

xy² = 1000

By division of this equations: x/y = 1; y = x

x³ = 1000, x = y = 10

(10, 10) is critical point for function D(x, y) = 2xy + 2000/x + 20000/y

iii)∂²D/∂x² = 4000/x³ = 4, ∂²D/∂y² = 4000/y³ = 4, ∂²D/∂x∂y = 2

∂²D/∂x²· ∂²D/∂y² - (∂²D/∂x∂y)² = 4·4 - 2² = 12 > 0.

So, function D has minimum at this critical point : D_min = D(10,10) = 200 + 200 + 200 = 600