To solve the optimization problem described, we first need to identify and formulate the objective function and constraints mathematically. The goal is to maximize \(9x_0 + 2x_1\), where \(x_0\) represents milligrams of vitamin K and \(x_1\) represents milligrams of vitamin A.

Given constraints are:
- Cardiovascular support index: \(2x_0 + 15x_1 \geq 62\) and \(2x_0 + 15x_1 \leq 82\)
- Kidney support index: \(7x_0 + 11x_1 \geq 74\) and \(7x_0 + 11x_1 \leq 111\)
- Immune support index: \(5x_0 + 8x_1 \geq 11\) and \(5x_0 + 8x_1 \leq 49\)
- Cognitive performance index: \(2x_0 + 8x_1 \geq 34\) and \(2x_0 + 8x_1 \leq 98\)
- Additional constraint: \(-4x_0 + 5x_1 \geq 0\)

Since both variables can be non-integer, we treat them as continuous variables.

Here is the Gurobi code to solve this problem:

```python
from gurobipy import *

# Create a new model
m = Model("Vitamin_Optimization")

# Add variables
x0 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_K")
x1 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_A")

# Set the objective function
m.setObjective(9*x0 + 2*x1, GRB.MAXIMIZE)

# Add constraints
m.addConstr(2*x0 + 15*x1 >= 62, name="cardiovascular_support_min")
m.addConstr(2*x0 + 15*x1 <= 82, name="cardiovascular_support_max")
m.addConstr(7*x0 + 11*x1 >= 74, name="kidney_support_min")
m.addConstr(7*x0 + 11*x1 <= 111, name="kidney_support_max")
m.addConstr(5*x0 + 8*x1 >= 11, name="immune_support_min")
m.addConstr(5*x0 + 8*x1 <= 49, name="immune_support_max")
m.addConstr(2*x0 + 8*x1 >= 34, name="cognitive_performance_min")
m.addConstr(2*x0 + 8*x1 <= 98, name="cognitive_performance_max")
m.addConstr(-4*x0 + 5*x1 >= 0, name="additional_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of Vitamin K: {x0.x}")
    print(f"Milligrams of Vitamin A: {x1.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```