To solve the problem, we need to find the absolute maximum and minimum of the function $f(x, y) = x^2 + y^2$ subject to the constraint $x^4 + y^4 = 16$.

Steps to Solve:

1. Set up the constraint: The constraint is $g(x, y) = x^4 + y^4 - 16 = 0$.

2. Use the method of Lagrange multipliers: Introduce a multiplier $\lambda$ and solve the equations: $\nabla f = \lambda \nabla g$ where $\nabla f = (2x, 2y)$ and $\nabla g = (4x^3, 4y^3)$.

   This gives: $2x = \lambda \cdot 4x^3, \quad 2y = \lambda \cdot 4y^3.$

   Simplify: \lambda = \frac{1}{2x^2} \quad \text{(if x \neq 0)}, \quad \lambda = \frac{1}{2y^2} \quad \text{(if y \neq 0)}.

3. Critical points: From symmetry and the constraint, we test $x = 0$ or $y = 0$, and also cases when $x = \pm y$. Substitute into the constraint $x^4 + y^4 = 16$ to find possible critical points.

4. Compute $f(x, y)$ at all critical points to find the maximum and minimum values.

Let me calculate these steps and provide the solution.from sympy import symbols, solve, Eq

Define variables

x, y, lambd = symbols('x y lambd')

Function and constraint

f = x2 + y2 # The function g = x4 + y4 - 16 # The constraint

Gradients for Lagrange multipliers

fx = 2 * x fy = 2 * y gx = 4 * x3 gy = 4 * y3

Solve using Lagrange multipliers

eq1 = Eq(fx, lambd * gx) eq2 = Eq(fy, lambd * gy) eq3 = Eq(x4 + y4, 16)

Solve for critical points

critical_points = solve([eq1, eq2, eq3], (x, y, lambd), dict=True)

Evaluate f(x, y) at critical points

results = [] for point in critical_points: x_val = point[x] y_val = point[y] results.append({'x': x_val, 'y': y_val, 'f(x,y)': f.subs({x: x_val, y: y_val})})

Find maximum and minimum values

results_sorted = sorted(results, key=lambda r: r['f(x,y)']) # Sort by f(x, y) results_sorted[{'x': -2**(3/4)I, 'y': -2*(3/4)I, 'f(x,y)': -4sqrt(2)}, {'x': -2**(3/4)I, 'y': 2*(3/4)I, 'f(x,y)': -4sqrt(2)}, {'x': 2**(3/4)I, 'y': -2*(3/4)I, 'f(x,y)': -4sqrt(2)}, {'x': 2**(3/4)I, 'y': 2*(3/4)I, 'f(x,y)': -4sqrt(2)}, {'x': 0, 'y': -2I, 'f(x,y)': -4}, {'x': 0, 'y': 2I, 'f(x,y)': -4}, {'x': -2I, 'y': 0, 'f(x,y)': -4}, {'x': 2I, 'y': 0, 'f(x,y)': -4}, {'x': -2, 'y': 0, 'f(x,y)': 4}, {'x': 0, 'y': -2, 'f(x,y)': 4}, {'x': 0, 'y': 2, 'f(x,y)': 4}, {'x': 2, 'y': 0, 'f(x,y)': 4}, {'x': -2**(3/4), 'y': -2**(3/4), 'f(x,y)': 4sqrt(2)}, {'x': -2**(3/4), 'y': 2**(3/4), 'f(x,y)': 4sqrt(2)}, {'x': 2**(3/4), 'y': -2**(3/4), 'f(x,y)': 4sqrt(2)}, {'x': 2**(3/4), 'y': 2**(3/4), 'f(x,y)': 4sqrt(2)}][{'x': -2**(3/4)I, 'y': -2*(3/4)I, 'f(x,y)': -4sqrt(2)}, {'x': -2**(3/4)I, 'y': 2*(3/4)I, 'f(x,y)': -4sqrt(2)}, {'x': 2**(3/4)I, 'y': -2*(3/4)I, 'f(x,y)': -4sqrt(2)}, {'x': 2**(3/4)I, 'y': 2*(3/4)I, 'f(x,y)': -4sqrt(2)}, {'x': 0, 'y': -2I, 'f(x,y)': -4}, {'x': 0, 'y': 2I, 'f(x,y)': -4}, {'x': -2I, 'y': 0, 'f(x,y)': -4}, {'x': 2I, 'y': 0, 'f(x,y)': -4}, {'x': -2, 'y': 0, 'f(x,y)': 4}, {'x': 0, 'y': -2, 'f(x,y)': 4}, {'x': 0, 'y': 2, 'f(x,y)': 4}, {'x': 2, 'y': 0, 'f(x,y)': 4}, {'x': -2**(3/4), 'y': -2**(3/4), 'f(x,y)': 4sqrt(2)}, {'x': -2**(3/4), 'y': 2**(3/4), 'f(x,y)': 4sqrt(2)}, {'x': 2**(3/4), 'y': -2**(3/4), 'f(x,y)': 4sqrt(2)}, {'x': 2**(3/4), 'y': 2**(3/4), 'f(x,y)': 4sqrt(2)}]success