Question 1199332
**1. Define the Function and its Gradient**

* **Function:** 
   f(x, y) = 6xy - (19x² + 3y²) - 36x - 14y + 13

* **Gradient of the Function:** 
   ∇f(x, y) = (∂f/∂x, ∂f/∂y) = (6y - 38x - 36, 6x - 6y - 14)

**2. Implement Gradient Ascent**

* **Initialize:**
   - `x0 = np.array([-3, -4])` 
   - `learning_rate = 0.01` 
   - `max_iter = 1000` 
   - `tol = 1e-6` 

* **Iterate:**
   1. Calculate the gradient at the current point `x`.
   2. Update `x` using the gradient ascent update rule: 
      `x = x + learning_rate * gradient`
   3. Check for convergence (e.g., if the magnitude of the gradient is below the tolerance).

**3. Find the Maximum**

* Run the gradient ascent algorithm.
* The final value of `x` after convergence will be the approximate location of the maximum.
* Evaluate the function `f(x)` at this point to find the maximum value.

**4. Adjust Learning Rate (if necessary)**

* If the algorithm doesn't converge or oscillates, try adjusting the `learning_rate`. 
    * A smaller learning rate can help with convergence but may slow down the process.
    * A larger learning rate can speed up convergence but may cause the algorithm to overshoot the maximum.

**Python Implementation**

```python
import numpy as np

def gradient_ascent(f, grad_f, x0, learning_rate, max_iter=1000, tol=1e-6):
  """
  Performs gradient ascent to find the maximum of a function.

  Args:
    f: The function to optimize.
    grad_f: The gradient of the function.
    x0: The initial point.
    learning_rate: The step size for the gradient ascent.
    max_iter: The maximum number of iterations.
    tol: The tolerance for convergence.

  Returns:
    x_opt: The optimal point found by the algorithm.
    f_opt: The maximum value of the function at x_opt.
  """

  x = np.array(x0)
  for _ in range(max_iter):
    gradient = grad_f(x)
    x = x + learning_rate * gradient
    if np.linalg.norm(gradient) < tol:
      break

  return x, f(x)

# Define the function
def f(x):
  return 6*x[0]*x[1] - (19*x[0]**2 + 3*x[1]**2) - 36*x[0] - 14*x[1] + 13

# Define the gradient of the function
def grad_f(x):
  return np.array([6*x[1] - 38*x[0] - 36, 6*x[0] - 6*x[1] - 14])

# Initial point
x0 = np.array([-3, -4])

# Learning rate
learning_rate = 0.01

# Perform gradient ascent
x_opt, f_opt = gradient_ascent(f, grad_f, x0, learning_rate)

# Print the results
print(f"Maximum point: {x_opt}")
print(f"Maximum value: {f_opt}") 
```

**Output:**

```
Maximum point: [-1.56250003 -3.89583352]
Maximum value: 68.39583333333324
```

**Therefore:**

* **x_opt = (-1.5625, -3.8958)** 
* **f_opt = 68.3958**

This result indicates that the maximum of the function f(x, y) is approximately 68.3958, which occurs at the point (-1.5625, -3.8958).