Understanding Local Extrema with Two Variables and Partial Derivatives

In summary, the conversation discusses finding and classifying local extrema using partial derivatives. The speaker initially struggles with solving for points with two variables, but is reminded to use the gradient of f and solve a system of equations. The solutions are the stationary points, which may be extrema but need to be checked. The conversation concludes with the speaker understanding the concept.
  • #1
Guero
15
0
we're doing partial derivatives, and i thought i understood them until:

[tex]f(x,y)=-x^2-y^2-10xy+4y-4x+2 [/tex]

we are meant to find and classify all local extrema. i got:

[tex]f_x=-2x-10y-4 [/tex] and [tex] f_y=-2y-10x+4[/tex]

i've never had two variables at this stage, and i can't solve for any of the points. the text doesn't seem to say anything about this...
 
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  • #2
yay! I'm now a level 30 thumb twiddler
 
  • #3
You solve [tex]\nabla f = \vec0[/tex] (gradient of f) which basically gives you a system of 2 equations here. The solutions are the 'stationary points' which may be extrema, but you have to check that (e.g. with a test)

So you solve:

[tex]\left\{ \begin{gathered}
\frac{{\delta f}}
{{\delta x}} = 0 \hfill \\
\frac{{\delta f}}
{{\delta y}} = 0 \hfill \\
\end{gathered} \right.[/tex]
 
  • #4
mm, i get: [tex]10y=-2x-4 [/tex] and [tex]10x=-2y+4 [/tex] for each of which there are infinite solutions for x and y. i thought that if i combined the equations (no reason) it might do it, but i still got [tex]y=x-1 [/tex]
 
  • #5
Yes, you have to combine them in a system.

[tex]\left\{ \begin{gathered}
2x + 10y + 4 = 0 \hfill \\
10x + 2y - 4 = 0 \hfill \\
\end{gathered} \right.[/tex]

Solving it should give [itex]\left( {\frac{1}
{2}, - \frac{1}
{2}} \right)[/itex]

However, although it's a stationary point, it's not an extremum.
 
  • #6
aha! thanks, i get it now
 

1. What is a local extremum with two variables?

A local extremum with two variables refers to a point on a two-dimensional graph where the function reaches either a maximum or minimum value within a small surrounding region.

2. How is a local extremum with two variables different from a global extremum?

A local extremum is a point where the function reaches a maximum or minimum value within a small area, while a global extremum is the highest or lowest point of the entire function.

3. How can I determine the location of a local extremum with two variables?

To determine the location of a local extremum, you can take the partial derivatives of the function with respect to each variable and set them equal to zero. Then, solve the resulting system of equations to find the coordinates of the local extremum.

4. Can a function have more than one local extremum with two variables?

Yes, a function can have multiple local extrema with two variables. These points can be either maximum or minimum values and can be located at different coordinates on the graph.

5. Are local extrema with two variables important in real-world applications?

Yes, local extrema with two variables are important in many real-world applications, such as optimization problems in economics, engineering, and physics. They can also be used to analyze and predict the behavior of complex systems.

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