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Math  /  Calculus

QuestionQuestion 2 1 pts
Jane is working on classifying the critical point a∈R4\boldsymbol{a} \in \mathbb{R}^{4} of a C3C^{3} function f:R4→Rf: \mathbb{R}^{4} \rightarrow \mathbb{R} using its Hessian matrix. The Hessian matrix is given by Hf(a)=[20000−30000000007]H f(a)=\left[\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 7 \end{array}\right]
What can she conclude by the second derivative test? The critical point is a local maximum. The critical point is a local minimum. The critical point is a saddle point. None of the above. The second derivative test is inconclusive.

Studdy Solution

STEP 1

1. The function f f is C3 C^3 , meaning it has continuous third derivatives.
2. The Hessian matrix Hf(a) Hf(a) is used to classify the critical point a \boldsymbol{a} .
3. The Hessian matrix is given as a diagonal matrix.

STEP 2

1. Understand the second derivative test for functions of multiple variables.
2. Analyze the given Hessian matrix.
3. Classify the critical point using the second derivative test.

STEP 3

The second derivative test for functions of multiple variables involves analyzing the eigenvalues of the Hessian matrix at the critical point.
For a function f:Rn→R f : \mathbb{R}^n \rightarrow \mathbb{R} , if the Hessian matrix Hf(a) Hf(a) at a critical point a \boldsymbol{a} is positive definite, then a \boldsymbol{a} is a local minimum. If it is negative definite, then a \boldsymbol{a} is a local maximum. If the Hessian has both positive and negative eigenvalues, a \boldsymbol{a} is a saddle point. If the Hessian is indefinite, the test is inconclusive.

STEP 4

The given Hessian matrix is:
Hf(a)=[20000−30000000007]Hf(a) = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 7 \end{bmatrix}
The eigenvalues of this diagonal matrix are the diagonal elements: 2,−3,0, 2, -3, 0, and 7 7 .

STEP 5

Since the Hessian matrix has both positive and negative eigenvalues, the critical point a \boldsymbol{a} is a saddle point.
The critical point is a saddle point.
The critical point is a saddle point. \boxed{\text{The critical point is a saddle point.}}

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