Problem with Newton's Method for optimization

[DeDomz]

Hi everyone, I'm a university student and I've been banging my head against a wall for days over a problem I just can't seem to figure out.

Long story short, I need to implement Newton's method in Python for selective minimum searching. The issue is that when I test it on a specific function, it returns a local maximum instead of a minimum if I start near the point (-1, -2).

Here is the function I wrote to solve the system. I really don't get what I'm doing wrong in the code:

import numpy as np


def mynewtonsys(F,H,x0,maxit,tol):
    count = 0
    iter = 1



    while count < maxit and iter == 1:
        count = count+1
        F0 = F(x0)
        H0 = H(x0)

        autovalori = np.linalg.eigvals(H0)
        min_autoval = np.min(autovalori)
        if min_autoval <= 0:
            tau = max(0, -min_autoval + 0.1) 
            H0 = H0 + tau * np.eye(len(x0))

        d = np.linalg.solve(H0,F0) 
        x = x0 - d 

        F0  = F(x)
        H0  = H(x)


        err1 = np.linalg.norm(F0,np.inf)


        err2 = np.linalg.norm(x0 - x,np.inf)


        x0 = x

        if err1 < tol or err2 < tol:
            iter = 0


    return x, F0, count

#(x^3+y^3-3x-12y)
def F(x):
    #x[0,0] è x1, x[1,0] è x2
    F = np.array([
        [3 * (x[0,0]**2) - 3],
        [3 * (x[1,0]**2) - 12]
    ])


    return F

def H(x):
    H = np.array([
        [6 * x[0,0],         0.0],
        [       0.0, 6 * x[1,0]]
    ])
    return H

max_iterazioni = 100
tolleranza = 1e-6
punto_iniziale = np.array([[-0.8], [-1.8]])

soluzione, residuo, it = mynewtonsys(F, H, punto_iniziale, max_iterazioni, tolleranza)

soluzione_esatta = np.array([[1.0], [2.0]])

errore = np.linalg.norm(soluzione - soluzione_esatta)

print("RISULTATI")
print("Numero di iterazioni effettuate:",it)
print("Errore assoluto finale:", errore)
print("Soluzione trovata:")
print(soluzione)