To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints algebraically.

Let's define:
- \(x_1\) as the amount of drug A (in mg) to be used in the mixture.
- \(x_2\) as the amount of drug B (in mg) to be used in the mixture.

The objective is to minimize the cost of the mixture. Given that it costs $0.50 per mg to purchase drug A and $0.30 per mg to purchase drug B, the objective function can be represented algebraically as:
\[ \text{Minimize: } 0.5x_1 + 0.3x_2 \]

The constraints are:
1. The mixture must contain at least 5 units of pain killer.
2. The mixture must contain at least 12 units of fever reliever.

Given that drug A contains 3 units/mg of pain killer and 2.5 units/mg of fever reliever, and drug B contains 2 units/mg of pain killer and 3.5 units/mg of fever reliever, we can formulate the constraints as follows:
1. \(3x_1 + 2x_2 \geq 5\) (pain killer constraint)
2. \(2.5x_1 + 3.5x_2 \geq 12\) (fever reliever constraint)

Additionally, since we cannot have a negative amount of drugs, we also have:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Therefore, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'amount of drug A in mg'), ('x2', 'amount of drug B in mg')],
    'objective_function': '0.5*x1 + 0.3*x2',
    'constraints': [
        '3*x1 + 2*x2 >= 5',
        '2.5*x1 + 3.5*x2 >= 12',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

Now, let's implement this problem using Gurobi in Python:
```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(name="drug_a", lb=0)
x2 = m.addVar(name="drug_b", lb=0)

# Set the objective function
m.setObjective(0.5*x1 + 0.3*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(3*x1 + 2*x2 >= 5, name="pain_killer_constraint")
m.addConstr(2.5*x1 + 3.5*x2 >= 12, name="fever_reliever_constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Amount of Drug A: {x1.x} mg")
    print(f"Amount of Drug B: {x2.x} mg")
    print(f"Total Cost: ${0.5*x1.x + 0.3*x2.x:.2f}")
else:
    print("No optimal solution found")
```