# /// script
# requires-python = ">=3.11"
# dependencies = ["sympy"]
# ///

import sympy as sp

def solve():
    x = sp.symbols('x', real=True)
    y = -2*x**2 + 8*x - 3
    
    # Find derivative to find critical points
    dy_dx = sp.diff(y, x)
    critical_points = sp.solve(dy_dx, x)
    
    print(f"Critical points (x): {critical_points}")
    
    if not critical_points:
        print("No critical points found.")
        return

    vertex_x = critical_points[0]
    vertex_y = y.subs(x, vertex_x)
    
    print(f"Vertex: ({vertex_x}, {vertex_y})")
    
    # Check second derivative to confirm maximum
    d2y_dx2 = sp.diff(dy_dx, x)
    print(f"Second derivative: {d2y_dx2}")
    
    if d2y_dx2 < 0:
        print("The vertex is a maximum.")
        max_val = vertex_y
    else:
        print("The vertex is a minimum.")
        max_val = vertex_y 

    print(f"Maximum value: {max_val}")

if __name__ == "__main__":
    solve()