Question 1194381: Given the following function in two variables x and y
f(x, y) = x^3y+2x^4+y^5
a. Find the Jacobean determinant at point P (3,2)
b. Find the Hessian determinant at point P (1,3)
Answer by parmen(42)   (Show Source):
You can put this solution on YOUR website! **a. Find the Jacobian Determinant at point P (3, 2)**
1. **Calculate Partial Derivatives:**
- ∂f/∂x = 3x²y + 8x³
- ∂f/∂y = x³ + 5y⁴
2. **Evaluate Partial Derivatives at P (3, 2):**
- ∂f/∂x(3, 2) = 3(3)²(2) + 8(3)³ = 54 + 216 = 270
- ∂f/∂y(3, 2) = (3)³ + 5(2)⁴ = 27 + 80 = 107
3. **Construct the Jacobian Matrix:**
- The Jacobian matrix is a 1x2 matrix:
[ ∂f/∂x ∂f/∂y ]
[ 270 107 ]
4. **Calculate the Jacobian Determinant:**
- Since it's a 1x2 matrix, the determinant is not defined. The Jacobian determinant is only defined for square matrices (where the number of rows equals the number of columns).
**b. Find the Hessian Determinant at point P (1, 3)**
1. **Calculate Second-Order Partial Derivatives:**
- ∂²f/∂x² = 6xy + 24x²
- ∂²f/∂y² = 20y³
- ∂²f/∂x∂y = 3x²
- ∂²f/∂y∂x = 3x²
2. **Evaluate Second-Order Partial Derivatives at P (1, 3):**
- ∂²f/∂x²(1, 3) = 6(1)(3) + 24(1)² = 18 + 24 = 42
- ∂²f/∂y²(1, 3) = 20(3)³ = 540
- ∂²f/∂x∂y(1, 3) = 3(1)² = 3
- ∂²f/∂y∂x(1, 3) = 3(1)² = 3
3. **Construct the Hessian Matrix:**
- The Hessian matrix is a 2x2 matrix:
[ ∂²f/∂x² ∂²f/∂x∂y ]
[ ∂²f/∂y∂x ∂²f/∂y² ]
[ 42 3 ]
[ 3 540 ]
4. **Calculate the Hessian Determinant:**
- det(Hessian) = (42)(540) - (3)(3) = 22680 - 9 = 22671
**Therefore:**
* The Jacobian determinant at point P (3, 2) is not defined.
* The Hessian determinant at point P (1, 3) is 22671.
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