The equation $f(x, y) = x^2 - 3y + 30$ needs to be analyzed for its turning points, minimum, maximum, or saddle points using the Hessian matrix. Let’s follow the steps systematically:

1. First-Order Partial Derivatives

The first-order partial derivatives are: $f_x = \frac{\partial f}{\partial x} = 2x, \quad f_y = \frac{\partial f}{\partial y} = -3.$

The critical points occur where $f_x = 0$ and $f_y = 0$: 2x = 0 \quad \Rightarrow \quad x = 0, \quad -3 = 0 \quad \Rightarrow \quad \text{No solution for y}. Thus, the function has no critical points because $f_y = -3$ is constant and cannot equal zero.

2. Second-Order Partial Derivatives (Hessian Matrix Components)

The second-order partial derivatives are:

f_{yy} = \frac{\partial^2 f}{\partial y^2} = 0, \quad f_{xy} = f_{yx} = 0.$$ The Hessian matrix is: $$H(x, y) = \begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}.$$ ### 3. Determinant of the Hessian The determinant of $$H(x, y)$$ is: $$\text{det}(H) = f_{xx}f_{yy} - (f_{xy})^2 = (2)(0) - (0)^2 = 0.$$ ### 4. Analysis of the Results - Since $$\text{det}(H) = 0$$, the Hessian test is inconclusive. This implies the nature of the point is indeterminate using only the Hessian determinant. - However, based on $$f_{xx} = 2 > 0$$, the function is convex in the $$x$$-direction, and no critical points satisfy both $$f_x = 0$$ and $$f_y = 0$$. Thus, we cannot identify a specific turning point. ### Conclusion The function $$f(x, y) = x^2 - 3y + 30$$ has **no critical points**. The absence of critical points precludes the identification of a minimum, maximum, or saddle point. --- Would you like further clarification or a deeper dive into the theoretical aspects? Here are some follow-up questions: 1. How is the determinant of the Hessian matrix related to the classification of critical points? 2. Why does the constant term $$-3y$$ prevent the existence of critical points in this case? 3. Can you provide another example where a Hessian determinant test is inconclusive? 4. How would the analysis change if $$f_y$$ were a function of $$y$$ instead of constant? 5. What does the convexity in the $$x$$-direction imply about the surface? **Tip:** If the Hessian determinant is zero, check the function's behavior in the region around the potential critical point for additional insights.