Second derivative test for functions of 2 variables

In summary, the conversation discussed using the second derivative test for functions of 2 variables to classify points as local maximum, local minimum, or inconclusive. The Hessian matrix is used to determine this, with the eigenvalues of the matrix being the key factor. The points (0,1) and (-1,1) were considered as examples, with (0,1) being classified as a maximum and (-1,1) being inconclusive.
  • #1
funzsquare
9
0
urgent! second derivative test for functions of 2 variables

Homework Statement


f(x,y)=x^4 - y^2 - 2x^2 + 2y - 7


Homework Equations


classify points (0,1) and (-1,1) as local maximum, local minimum or inclusive


The Attempt at a Solution



f(x,0)=4x^3 - 0 - 4x + 0 - 0 = 4x^3-4x
f'(x,0)=12x^2-4
f''(x,0)=24x

f(0,y)=0 - 2y - 0 + 2 - 0 = -2y+2
f'(0,y)=-2
f''(0,y)=0

how to find the points as above? as i am stuck.
 
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  • #2


Why are you fixing x=0 and y=0 when you calculate the derivates? You need to use the Hessian matrix to solve your problem.
 
  • #3


so i use the 2x2 matrix?
do i equate the 4x^3-4x=0 & -2y+2=0?

sry as i am new just start learning.
 
  • #4


No worries. Slow down though. Why are you equating anything to zero? Can you tell me what the definition of the Hessian is? If you can, calculate the components and put them into the matrix. After that, tell me how we can use the Hessian to determine whether a point is maximal, minimal, or a saddle?
 
  • #5


i thought of finding the stationary point.
square matrix of second-order partial derivatives.
fx=4x^3-4x
fy=-2y+2
is this correct?
 
  • #6


Yep. But now you need fxx fxy and fyy.
 
  • #7


You can find the stationary points if you want, but I believe they've already been given to you.
 
  • #8


fxx=12x^2-4
fyy=-2
fyx=0
fxy=0

correct?
 
Last edited:
  • #9


Throw a negative sign in front of fyy and that will be correct. Now, how can you determine whether or not (0,1) or (-1,1) are min/max/saddle using the Hessian?
 
  • #10


[12x^2-4 0]
-2 0

do i need to find the eigenvalues?
 
  • #11


0,1 inconclusive -1,1 maximum?
 
  • #12


anyone?
 
  • #13


up!
 
  • #14


Check your matrix again, it should be

[tex] \begin{pmatrix} 12x^2 & 0 \\ 0 & -2 \end{pmatrix} [/tex]

Now you do indeed have to find the eigenvalues of this matrix for each point you want to consider. However, the eigenvalues of diagonal matrices are quite easy...
 
  • #15


so 0,1 maximum -1,1 inconclusive
 

Related to Second derivative test for functions of 2 variables

What is the second derivative test for functions of 2 variables?

The second derivative test is a mathematical tool used to determine the nature of a critical point in a function of 2 variables. It involves calculating the second derivatives of the function and using their signs to determine whether the critical point is a maximum, minimum, or saddle point.

How do you find the second derivatives of a function of 2 variables?

To find the second derivatives of a function of 2 variables, you first need to find the first derivatives with respect to both variables. Then, you take the partial derivatives of each first derivative with respect to the corresponding variable. These partial derivatives are the second derivatives.

What does the sign of the second derivatives indicate in the second derivative test?

The sign of the second derivatives indicates the concavity of the function at the critical point. A positive second derivative means the function is concave up, indicating a minimum point. A negative second derivative means the function is concave down, indicating a maximum point. A second derivative of zero means the test is inconclusive and further analysis is needed.

What are the steps to perform the second derivative test?

The steps to perform the second derivative test are as follows:

  1. Find the first derivatives of the function with respect to both variables.
  2. Take the partial derivatives of each first derivative with respect to the corresponding variable to find the second derivatives.
  3. Evaluate the second derivatives at the critical point.
  4. Determine the sign of the second derivatives.
  5. Based on the sign of the second derivatives, determine the nature of the critical point (maximum, minimum, or saddle point).

What are some real-world applications of the second derivative test for functions of 2 variables?

The second derivative test has various real-world applications in fields such as economics, engineering, and physics. Some examples include:

  • Optimization of production processes in manufacturing or agriculture.
  • Maximizing profits in business by finding the optimal price for a product.
  • Designing efficient transportation routes by finding the shortest path between two points.
  • Studying the stability of a physical system by analyzing its critical points.

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