Example Questions
Example Question #25 :The Hessian
Use the Hessian matrix, if applicable, to answer this question:
Does the graph ofhave a local maximum, a local minimum, or a saddle point at?
The graph ofhas a local maximum at.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
The graph ofdoes not have a critical point at.
The graph ofhas a saddle point at.
The graph ofhas a local minimum at.
The graph ofhas a saddle point at.
First, it must be established that the graph ofhas a critical point at; this holds if, so the first partial derivatives ofmust be evaluated at:
The graph ofhas a critical point at, so the Hessian matrix test applies.
The Hessian matrixis the matrix of partial second derivatives
,
the determinant of which can be used to determine whether a critical point ofis a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of:
All four partial second derivatives are constant; the Hessian matrix atis
Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product:
The determinant of the Hessian is negative, so the graph ofhas a saddle point at.
Example Question #26 :The Hessian
Use the Hessian matrix, if applicable, to answer this question:
Does the graph ofhave a local maximum, a local minimum, or a saddle point at?
The graph ofhas a local minimum at.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
The graph ofhas a saddle point at.
The graph ofhas a local maximum at.
The graph ofdoes not have a critical point at.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
First, it must be established that the graph ofhas a critical point at; this holds if, so the first partial derivatives ofmust be evaluated at:
The graph ofhas a critical point at, so the Hessian matrix test applies.
The Hessian matrixis the matrix of partial second derivatives
,
the determinant of which can be used to determine whether a critical point ofis a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of:
The Hessian matrix atis
Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product:
Since the determinant of the Hessian is 0, the Hessian matrix test is inconclusive.
Example Question #27 :The Hessian
Use the Hessian matrix, if applicable, to answer this question:
Does the graph ofhave a local maximum, a local minimum, or a saddle point at?
The graph ofhas a saddle point at.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
The graph ofhas a local minimum at.
The graph ofdoes not have a critical point at.
The graph ofhas a local maximum at.
The graph ofhas a local maximum at.
First, it must be established that the graph ofhas a critical point at; this holds if, so the first partial derivatives ofmust be evaluated at:
The graph ofhas a critical point at, so the Hessian matrix test applies.
The Hessian matrixis the matrix of partial second derivatives
,
the determinant of which can be used to determine whether a critical point ofis a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of:
The Hessian matrix atis
Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product:
The determinant is positive, makinga local extremum. Sinceis negative,is a local maximum.
Example Question #28 :The Hessian
Use the Hessian matrix, if applicable, to answer this question:
Does the graph ofhave a local maximum, a local minimum, or a saddle point at?
The graph ofdoes not have a critical point at.
The graph ofhas a saddle point at.
The graph ofhas a local maximum at.
The graph ofhas a local minimum at.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
First, it must be established that the graph ofhas a critical point at; this holds if, so the first partial derivatives ofmust be evaluated at:
The graph ofhas a critical point at, so the Hessian matrix test applies.
The Hessian matrixis the matrix of partial second derivatives
,
the determinant of which can be used to determine whether a critical point ofis a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of:
All four partial second derivatives are constants. The Hessian matrix at any point, including, is
;
Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product:
Since the determinant of the Hessian is 0, the Hessian matrix test is inconclusive.
Example Question #29 :The Hessian
Consider the function.
Determine whether the graph of the function has a critical point at; if so, use the Hessian matrixto identifyas a local maximum, a local minimum, or a saddle point.
The graph ofhas a local maximum at.
The graph ofhas a local minimum at.
The graph ofhas a saddle point at.
The graph ofdoes not have a critical point at.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
The graph ofhas a local maximum at.
First, it must be established thatis a critical point of the graph of; this holds if and only if both first partial derivatives are equal to 0 at this point. Find the partial derivatives and evaluate them at:
Thus, the graph ofhas a critical point at.
The Hessian matrix is the matrix of partial second derivatives
;
Find these derivatives and evaluate them at:
At, the Hessian matrix is
The determinant of this matrix is
Since the determinant of the Hessian matrix is positive, the graph ofhas a local extremum at; since, a negative value, it is a local maximum.
Example Question #30 :The Hessian
Consider the function.
Determine whether the graph of the function has a critical point at; if so, use the Hessian matrixto identifyas a local maximum, a local minimum, or a saddle point.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
The graph ofhas a saddle point at.
The graph ofhas a local maximum at.
The graph ofhas a local minimum at.
The graph ofdoes not have a critical point at.
The graph ofhas a critical point at, but the Hessian matrix test is inconclusive.
First, it must be established thatis a critical point of the graph of; this holds if and only if both first partial derivatives are equal to 0 at this point. Find the partial derivatives and evaluate them at:
Thus, the graph ofhas a critical point at.
The Hessian matrix is the matrix of partial second derivatives
;
Find these derivatives and evaluate them at:
The Hessian matrix, evaluated at, ends up being the matrix. The determinant of the matrix is 0, which means that the Hessian matrix test is inconclusive.
Example Question #31 :The Hessian
Let
Which of the following doesnotappear in the Hessian matrix of?
The Hessian matrix ofis the matrix of partial second derivatives
To identify which choice does not give an entry of the matrix, we need to find all nine partial derivatives; however, since,, and, we need only find six such derivatives. They are as follows:
Of the five given choices, onlyis not one of the partial second derivatives. This is the correct choice.
Example Question #32 :The Hessian
Let.
Find the Hessian matrix.
The Hessian matrix is the matrix of partial second derivatives
.
can be rewritten as
, then
This makes the partial second derivatives easier to find.
The Hessian matrix foris
Example Question #33 :The Hessian
Consider the function.
Which of the following expressions does not appeartwicein the Hessian matrix of?
The Hessian matrix is the matrix of partial second derivatives
.
Since, we only need to find six partial second derivatives and compare them to the five choices.
As stated before,
,
so all three expressions will appear twice in the Hessian matrix.
Also note that
,
so this expression appears twice as well.
,
however, only appears once. This is the correct choice.