2. Find FOCASOL of the following a) z = x5 - 3x2y + y6 b) z = x/ySee answer

Daftar Isi

2 Find FOCASOL of the following a z x5 3x2y y6 b z x y

Question

2. Find FOCASOL of the following
a) z = x⁵ – 3x²y + y⁶
b) z = x/y

Super Answer

I’ll help you find the First Order Conditions for Absolute Solution (FOCASOL) for both functions.

a) For z = x⁵ – 3x²y + y⁶

Step 1: Find Partial Derivative with respect to x (∂z/∂x)

∂z/∂x = 5x⁴ – 6xy = 0

Step 2: Find Partial Derivative with respect to y (∂z/∂y)

∂z/∂y = -3x² + 6y⁵ = 0

Step 3: Solve the System of Equations

From ∂z/∂y = 0:
-3x² + 6y⁵ = 0
3x² = 6y⁵
x² = 2y⁵

From ∂z/∂x = 0:
5x⁴ = 6xy
5x³ = 6y

Step 4: Substitute to Find Solution

Substitute x² = 2y⁵ into 5x³ = 6y:
5(2y⁵)^(3/2) = 6y
5(2√2y^(15/2)) = 6y
10√2y^(15/2) = 6y
y^(15/2) = (3/5√2)y
y = 0 is one solution

Final Answer for (a)

Critical point is (0,0)

b) For z = x/y

Step 1: Find Partial Derivative with respect to x (∂z/∂x)

∂z/∂x = 1/y = 0

Step 2: Find Partial Derivative with respect to y (∂z/∂y)

∂z/∂y = -x/y² = 0

Step 3: Analyze Equations

1/y = 0 implies y = ∞
-x/y² = 0 implies x = 0 or y = ∞

Final Answer for (b)

No finite solution exists as y cannot be infinity in real domain. This function has no critical points in the finite domain.