Finding Critical Points: Solving f_x & f_y

In summary, the conversation discusses finding the critical points for the partial derivatives of a given function. The first partial leads to two possibilities for x and the second equation becomes quadratic when these values are plugged in. The conversation also includes some humorous banter between the speakers, one of whom has changed their username.
  • #1
frasifrasi
276
0
I have to find the critical points for the partials:

f_x = y/3(24 - 12x - 4y) = 0
f_y = x/3(24 - 6x - 8y) = 0

I get y = 0 and x = (6-y)/3 for x in the first partial. How am I supposed to proceed? If I plug these into the secon, it gets nasty. Can anyone demonstrate how this is done?

Thank you.
 
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  • #2
First, you mean, e.g. f_x=(y/3)(24-12x-4y). Put enough parentheses into make it unambiguous. Second, you don't get y=0 AND x=(6-y)/3, you get y=0 OR x=(6-y)/3. Plug each possibility into the second equation. Third, it doesn't get 'nasty'. You get quadratic equations for x or y. That's not considered 'nasty'.
 
  • #3
That is some gruesome math man.
 
  • #4
Quadratics?? Are you serious?
 
  • #5
Dick said:
Quadratics?? Are you serious?
LOL, you just made my night :)
 
  • #6
Well, thanks. I am trying for the comedian of the year award. Aren't you the artist formerly known as rocophysics? Why the name change?
 

1. What is the purpose of finding critical points in a function?

The critical points of a function are important because they help us determine the maximum and minimum values of the function. This can be useful in optimization problems or in understanding the behavior of a function.

2. How do you find the critical points of a function?

To find the critical points, we need to solve for the partial derivatives of the function with respect to each variable (fx and fy). Then, we set these partial derivatives equal to 0 and solve for the values of x and y that satisfy the equations. These values will be the critical points of the function.

3. What does it mean if a function has no critical points?

If a function has no critical points, it means that there are no values of x and y that make the partial derivatives equal to 0. This could indicate that the function is constant, has no local extrema, or has a very complicated shape.

4. Can a function have multiple critical points?

Yes, a function can have multiple critical points. These points can represent local maximum or minimum values, as well as saddle points where the function changes direction.

5. How do critical points relate to the graph of a function?

The critical points of a function correspond to the points on the graph where the tangent line is horizontal (has a slope of 0). This can help us identify the shape and behavior of the function, such as where it reaches its maximum or minimum values.

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