Question 1194381
**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.