Find the stationary point of a function of two variables (2)

Error

There was an error loading the page.

Contributors

Bill Foster

Feedback

From users who are members of Bill's workspace :

Bill Foster

said

Ready to use

9 years, 2 months ago

History

○

Bill Foster 9 years, 2 months ago

Gave some feedback: Ready to use

○

Bill Foster 9 years, 2 months ago

Created this.

There is only one version of this question that you have access to.

There is one other version that you do not have access to.

Question statement

In the following question find $a$ such that $(0,a)$ is a critical point of the function:

$f(x,y)=\simplify[std]{{a}*x^3+{b}*x^2*y+{c}*y^2+{d}y+{f}}$

Also find the type of this critical point.

Give any introductory information the student needs.

			Name	Type	Generated Value

rawstatval	decimal	dec("-13.75")
lmin	string	Local minimum
q1	integer	3
dvalue	integer	-30
neither	string	Saddle point
b	integer	3
a	integer	2
c	integer	-1
lmax	string	Local maximum
p1	integer	3
d	integer	-5
f	integer	5
y	rational	5/2
x	integer	0
type	list	[ 0, 0, 1 ]
desc	string	$\displaystyle D \lt 0 \Righta

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
a*x^3+b*x^2*y+c*y^2+d*y+f

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `decimal`

dec("-13.75")

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
1. Step Information only
Part b) Gap-fill
1. Gap Gap 0. Matrix entry
  Add
2. Gap Gap 1. Choose one from a list
  Add
1. Step Information only

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$a=\;\;\;$ Gap 0.. Input as a fraction or an integer and not as a decimal.

Input the value of $f(x,y)$ at the critical point $(0,a)$ :

$f(0,a)=\;\;$ Gap 1.(Input to 2 decimal places).

Advice

a) The $(x,y)$ coordinates of the critical point of a function of 2 variables $f(x,y)$ are given by solving
the following 2 equations for $x$ and $y$ :

$\begin{eqnarray*} \frac{\partial f}{\partial x} &=&0\\ \\ \frac{\partial f}{\partial y}&=&0 \end{eqnarray*}$

In this case you get two equations to solve for $x$ and $y$ :

$\begin{eqnarray*} \simplify[std]{{3*a}x^2+{2*b}x*y}&=&0\\ \\ \simplify[std]{{b}x^2+{2*c}y+{d}}&=&0 \end{eqnarray*}$
The first equation factorises as

$\simplify{x*({3*a}x+{2*b}y)=0}$ and we find that either:

$x=0,\text{ or } \simplify[std]{y={-3*a}x/{2*b}}$

On substituting $x=0$ into the second equation we get $y=\simplify[std]{{-d}/2}$ .

Hence $(0,\simplify[std]{{-d}/2})$ is a critical point and it is the only critical point where $x=0$ .

On substituting these values for $x$ and $y$ into $f(x,y)$ we get:

$f\left(0,\simplify[std]{{-d}/2}\right) = \simplify[std]{{rawstatval}}.$

b) The Hessian at a point $(x,y)$ is:

$\left(\begin{array}{cc}\simplify[std]{{6*a}x+{2*b}y}&\simplify[std]{{2*b}x}\\ \simplify[std]{{2*b}x}&\var{2*c}\end{array}\right)$

Evaluated at the critical point $(0,\simplify[std]{{-d}/2})$ we obtain:

$\left(\begin{array}{cc}\simplify[std]{{-d*b}}&0\\ 0&\var{2*c}\end{array}\right)$

The determinant is $D=\var{dvalue}$ .

We see that on looking at the information given by Steps above we have:

$\var{desc}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Information only
Gap-fill
Gap-fill		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Information only

This question is used in the following exam:

Differentiation by Bill Foster in Bill's workspace.

Loading...

Error

Find the stationary point of a function of two variables (2)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 9 years, 2 months ago

Bill Foster 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `decimal`

Parts

Gap-fill

Loading...

Error

Find the stationary point of a function of two variables (2)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 9 years, 2 months ago

Bill Foster 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: decimal

Parts

Gap-fill

Generated value: `decimal`