Please help me with this, i need a detailed explaination please.

Question

Find the absolute maximum and minimum values of the function f(x, y) = x^2 + y^4 on D = {(x, y)|x^2 + y^2 ≤ 9}.

Answer 1

∂f/∂x = 2x = 0; ∂f/dy = 4y³ = 0; y = 0. (0, 0) is critical point inside D.

Now on the board: x² = 9 - y²; f(x, y) = 9 - y² + y⁴; df/dy = - 2y + 4y³; 2y(- 1 + 2y²) = 0; y = 0 or y = ±1/√2;

y = 0; x = ±3, so (3,0) and (-3, 0) are critical points

y = ±1/√2, x = ±√17/2 and we have 4 more critical points: (√17/2,1/√2); (-√17/2,1/√2), (√17/2,-1/√2 ),

(- √17/2, -1/√2√)

f(0,0) = 0 abs min; f(3,0) = 9 abs max; f(-3,0) = 9 abs max; f(√17/2,1/√2) = f(-√17/2,1/√2) = f(√17/2,-1/√2) = f(-√17/2,-1/√2) = 8.75