Find the local max and local min and saddle points of f(x, y) = x^4 + y^4 - 4xy +1
To find the local maxima, local minima, and saddle points of the function f(x, y) = x^4 + y^4 - 4xy + 1, we need to follow these steps:
1. Find the partial derivatives with respect to x and y:
∂f/∂x = 4x^3 - 4y
∂f/∂y = 4y^3 - 4x
2. Set both partial derivatives equal to zero to find the critical points:
4x^3 - 4y = 0 ⟹ x^3 = y
4y^3 - 4x = 0 ⟹ y^3 = x
3. Solve the system of equations formed in step 2 to find the critical points. Substituting y = x^3 into the second equation, we get:
(x^3)^3 = x ⟹ x^9 = x
From this equation, it is evident that x = 0 is one of the critical points.
4. To find more critical points, we can consider the behavior of the partial derivatives as x or y approaches infinity. As x or y becomes very large, the higher-order terms dominate the function, and the partial derivatives become approximately:
∂f/∂x ≈ 4x^3
∂f/∂y ≈ 4y^3
Since both partial derivatives become positive as x or y approaches infinity, there are no more critical points to examine.
Now, let's determine the nature of the critical point we found (x = 0, y = 0) and classify it as a local maximum, local minimum, or saddle point.
5. We need to investigate the behavior of the function in the vicinity of the critical point. To do this, we can use the second derivative test.
The second partial derivatives are:
∂²f/∂x² = 12x^2
∂²f/∂y² = 12y^2
∂²f/∂x∂y = -4
6. Plug in the values of x = 0 and y = 0 into the second partial derivatives:
∂²f/∂x²(0, 0) = 0
∂²f/∂y²(0, 0) = 0
∂²f/∂x∂y(0, 0) = -4
7. Calculate the discriminant (D) using the second partial derivatives:
D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)^2
D = (0)(0) - (-4)^2 = -16
8. Analyze the discriminant:
If D > 0 and (∂²f/∂x²)(∂²f/∂y²) > 0, then the critical point is a local minimum.
If D > 0 and (∂²f/∂x²)(∂²f/∂y²) < 0, then the critical point is a local maximum.
If D < 0, then the critical point is a saddle point.
In this case, D < 0, so the critical point (0, 0) is a saddle point.
Therefore, the function f(x, y) = x^4 + y^4 - 4xy + 1 has one saddle point at (0, 0).