How do I find the critical points of this multivariable function?

In summary, a critical point of a multivariable function is a point where all partial derivatives are equal to 0. To find these derivatives, the function is differentiated with respect to each variable while holding others constant. Critical points can exist at the boundary and the second derivative test can determine their nature. Multiple critical points are possible and should be analyzed for a complete understanding of the function.
  • #1
Inertigratus
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Homework Statement


The function is defined in the xy-plane. Find all critical points and determine their characteristics.


Homework Equations


f(x, y) = x4 + 4xy + y4


The Attempt at a Solution


I tried finding the partial derivatives.
f'x = 4x3 - 4y
f'y = 4y3 - 4x
Setting them to 0 gives y = x3.
If x = 1, y = 1, if x = 2, y = 8, etc.
This means that there's an infinite amount of critical points, this can't be right?
 
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  • #2
Nevermind, my brain is kinda tired... just realized that x = y^3 and y = x^3 is not the same equation. Well thanks anyway :).
 

FAQ: How do I find the critical points of this multivariable function?

1. What is a critical point?

A critical point of a multivariable function is a point where the partial derivatives of all the variables are equal to 0. In other words, it is a point where the function is neither increasing nor decreasing in any direction.

2. How do I find the partial derivatives of a multivariable function?

To find the partial derivatives of a multivariable function, you need to take the derivative of the function with respect to each variable while holding all other variables constant. This means treating all other variables as constants and using the rules of differentiation to find the derivatives.

3. Can critical points exist at the boundary of a multivariable function?

Yes, critical points can exist at the boundary of a multivariable function. In this case, the critical point is known as a boundary point. To find the critical points at the boundary, you may need to use additional techniques such as Lagrange multipliers.

4. How do I use the second derivative test to determine the nature of a critical point?

The second derivative test involves finding the second-order partial derivatives of the multivariable function at the critical point. If the second derivative is positive, the critical point is a local minimum. If the second derivative is negative, the critical point is a local maximum. If the second derivative is zero, the test is inconclusive and further analysis is needed.

5. Is it possible to have more than one critical point in a multivariable function?

Yes, it is possible to have multiple critical points in a multivariable function. These points can be local minima, local maxima, or saddle points. It is important to analyze all critical points to fully understand the behavior of the function.

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