Math Problem Statement
Solution
Let’s analyze the problem shown in your image. The task is to identify the critical points of the function and classify them.
Step-by-step Solution:
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Given function:
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Find the partial derivatives: The critical points occur when both partial derivatives and are zero.
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Partial derivative with respect to :
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Partial derivative with respect to :
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Set the partial derivatives equal to zero: Solve the system of equations:
From equation (1):
Substitute into equation (2):
This confirms works for all solutions. Hence, the critical point is:
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Classify the critical point: To classify the critical point, compute the second partial derivatives:
Compute the determinant of the Hessian matrix:
\begin{vmatrix} \frac{\partial^2 z}{\partial x^2} & \frac{\partial^2 z}{\partial x \partial y} \\ \frac{\partial^2 z}{\partial x \partial y} & \frac{\partial^2 z}{\partial y^2} \end{vmatrix} = \begin{vmatrix} 2 & -2 \\ -2 & 2 \end{vmatrix}$$ Compute $$H$$: $$H = (2)(2) - (-2)(-2) = 4 - 4 = 0$$ Since the determinant of the Hessian matrix is zero, the test is inconclusive. Further analysis would be required to classify the critical point.
Summary:
- The critical point is .
- The classification is inconclusive using the Hessian determinant.
Let me know if you'd like detailed explanations for any step or further classification using another method!
Questions to Explore:
- What is the significance of a critical point in multivariable calculus?
- How does the Hessian determinant help classify critical points?
- What happens when the determinant of the Hessian matrix is zero?
- Can this function have a saddle point at ? How would you determine this?
- How can eigenvalues of the Hessian matrix help classify critical points?
Tip:
When the Hessian determinant test fails (i.e., ), consider analyzing the function's behavior in different directions to classify the critical point.
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Critical Points
Hessian Matrix
Formulas
Partial derivative formulas: ∂z/∂x and ∂z/∂y
Hessian determinant formula: H = f_xx * f_yy - (f_xy)^2
Theorems
Critical Point Theorem
Hessian Determinant Test
Suitable Grade Level
College Level (Calculus 3 or equivalent)
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