Local Extrema with Partial d/dx

In summary, the conversation is about finding a local max/min using the second derivative test. The individual has already calculated the first and second partials and is unsure of how to use the second derivative test when there are no variables to plug in for the critical point. The expert explains that in this case, the second derivative test will result in a constant and there is no need to plug in any values. The conversation ends with the individual realizing their mistake in calculating the first derivative. The expert concludes by stating that when the second derivatives are constants, there can only be one critical point.
  • #1
Oglethorpe
5
0
Hello, I'm been stuck on this problem and I've been staring blankly at it way too long. I stumbled upon here and thought I'd ask for help? :P

Alright well, I'm looking for a local max/min, and I've already done the first partials and I got *f(x)=2x-y and f(y)=-x+2y+6; I'm sure those are right and the seconds partials were; f(xx)=2, f(xy)=-1, and f(yy)=2

...well, how do I use the Second derivative test on that? Where do I plug my critical point in (which I got (3/2, -6) is that right?)? Anyways, help would be great! :D

*It really means f subscript x, and so do the others, I just don't know how to use subscripts on a keyboard.
 
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  • #2
Just use '_' for subscript, as in f_x=2x-y. (3/2,-6) doesn't satisfy f_x=0, does it? Why do you think that would be a critical point?
 
  • #3
Oh thanks, and I simplified the f_x to y=2x-9, which I then substituted into f_y and went from f(y)=-x+2y+6 to -2x+9+4x-18+6=0, which simplifies to 2x-3=0 resulting in x=3/2. I'm more curious on how to take a second derivatives test when there are no variables to plug in a critical point, however.
 
  • #4
Oglethorpe said:
Oh thanks, and I simplified the f_x to y=2x-9, which I then substituted into f_y and went from f(y)=-x+2y+6 to -2x+9+4x-18+6=0, which simplifies to 2x-3=0 resulting in x=3/2. I'm more curious on how to take a second derivatives test when there are no variables to plug in a critical point, however.

Well, x=3/2 is still wrong for a critical point, but ok, the second derivative test just gives you a constant, yes? Is it positive or negative? There's no need to plug anything in in this case.
 
  • #5
Why would that be? And yes, I get

f_xx=2
f_xy=-1
f_yy=2
 
  • #6
Oglethorpe said:
Why would that be? And yes, I get

f_xx=2
f_xy=-1
f_yy=2

x=3/2 is wrong as a critical point. If you can tell me what you think the corresponding y is then it should be obvious. Unless you mistyped f_y or f_x.
 
  • #7
Oh! I typed f_x=2x-y when I should've typed f_x=2x-y-9
 
  • #8
The point is that your second derivatives are constant functions. You do not put in the values of x and y for the critical point anywhere. [itex]f_xxf_yy- fxy^2[/itex] is just (2)(2)- 1= 3> 0.

When the second derivatives are constants, the first derivatives must be linear so there can be only one critical point anyway.
 

FAQ: Local Extrema with Partial d/dx

1. What is a local extrema?

A local extrema is a point on a graph where the function reaches either a maximum or minimum value. This point is considered to be "local" because it is the highest or lowest point within a specific interval or range.

2. How do you find local extrema using partial derivatives?

To find local extrema using partial derivatives, you first need to take the derivative of the function with respect to the variable you are interested in. Then, set the derivative equal to zero and solve for the variable. This will give you the x-coordinate of the local extrema. To find the y-coordinate, plug the x-value into the original function.

3. What is the difference between a local extrema and a global extrema?

A global extrema is the highest or lowest point on the entire graph, while a local extrema is the highest or lowest point within a specific interval. A function can have multiple local extrema, but only one global extrema.

4. Can a function have more than one local extrema?

Yes, a function can have multiple local extrema. This can happen when the function has multiple peaks and valleys within a specific interval, and each of these points would be considered a local extrema.

5. How can you determine if a local extrema is a maximum or minimum?

To determine if a local extrema is a maximum or minimum, you can use the second derivative test. Take the second derivative of the function and plug in the x-value of the local extrema. If the result is positive, the point is a local minimum. If the result is negative, the point is a local maximum. If the second derivative is zero, the test is inconclusive.

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