How many critical points are there in a function with multiple variables?

In summary, there are 27 critical points for the function f(x,y,z) = x^4 + y^4 + z^4 - x^2 - y^2 - z^2, including the trivial solutions of all 0 and all \pm1, and the solutions x = 0 & \pm1/\sqrt{2}, y = 0 & \pm1/\sqrt{2}, z = 0 & \pm1/\sqrt{2}. Additional critical points can be found by setting x^2 = 1-\epsilon, y^2 = 1 + \epsilon, and z^2 = \epsilon, where \epsilon is in the range of [0,1].
  • #1
squenshl
479
4
I was given f(x,y,z) = x4 + y4 + z4 - x2 - y2 - z2.
I found that (or least I think it's these) x = 0 & [tex]\pm1/\sqrt{2}[/tex], y = 0 & [tex]\pm1/\sqrt{2}[/tex], z = 0 & [tex]\pm1/\sqrt{2}[/tex].
What I'm stuck with is exactly how much critical points are there, by the looks of things there are a few but I'm not too sure, how do I arrange them?
 
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  • #2
You have the trivial all 0 and all [itex]\pm1[/itex] solutions, and imagine
[tex]\begin{align*} x^2 &= 1-\epsilon,\\ y^2 &= 1 + \epsilon,\\ \epsilon&\in[0,1]\end{align*}[/tex]

Then what happens to

[itex]
f(x,y,z) = x^2(x^2 -1) + y^2(y^2-1) + z^2(z^2-1)
[/itex]
 
  • #3
Wow, 27 critical points.
 

1. What is a critical point in calculus?

A critical point in calculus is a point on a function where the derivative is either zero or undefined. This means that the slope of the function at that point is either flat or does not exist.

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

To find critical points of a function, you need to take the derivative of the function and set it equal to zero. Then, solve for the variable to determine the x-values of the critical points. You can also check for undefined points by setting the derivative equal to undefined and solving for the variable.

3. What is the significance of critical points?

Critical points help us analyze the behavior of a function. They can indicate where a function has a minimum or maximum value, as well as any potential inflection points. They are also used to find the optimal solutions to optimization problems.

4. Can a function have multiple critical points?

Yes, a function can have multiple critical points. These can occur at any point where the derivative is equal to zero or undefined. It is important to analyze all critical points to fully understand the behavior of a function.

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

The critical points of a function correspond to the points where the slope of the graph is either flat or undefined. This means that the graph will have a horizontal tangent line at these points. The behavior of the function around critical points can also indicate the presence of a minimum or maximum value.

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