Final answer:
To find the local maximum and minimum values and saddle points of the function f(x,y) = 2x^3 - 6x + 6xy^2, we need to find the critical points and use the second partial derivatives test. The function has a local minimum at (1,0) and a saddle point at (-1,0).
Step-by-step explanation:
To find the local maximum and minimum values and saddle points of the function f(x,y) = 2x^3 - 6x + 6xy^2, we need to find the critical points where the first derivatives are equal to zero. Differentiating f(x,y) with respect to x and y, we get:
∂f/∂x = 6x^2 - 6 + 6y^2 = 0
∂f/∂y = 12xy = 0
Solving these equations simultaneously, we find the critical points (x, y) as (1, 0), (-1, 0), and (0, 0).
To determine whether each critical point is a local maximum, local minimum or saddle point, we need to use the second partial derivatives test. Evaluating the second derivatives at each critical point:
∂^2f/∂x^2 = 12x, ∂^2f/∂y^2 = 12x, ∂^2f/∂x∂y = 12y
Substituting the critical points (x, y) into these second derivatives, we find that:
For (1, 0): ∂^2f/∂x^2 = 12 > 0 (local minimum)
For (-1, 0): ∂^2f/∂x^2 = -12 < 0 (saddle point)
For (0, 0): ∂^2f/∂x^2 = 0 (test inconclusive)
Therefore, the local maximum value is DNE, the local minimum value is f(1,0)=2, and the saddle point is at (-1,0).