Classifying Critical Points of Multivariable Functions

In summary, the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3 are (0,0,0) and any point where the gradient equals 0. The gradient can be calculated using the equations df/dx = y + z +3x^2, df/dy = x + z + 3y^2, and df/dz = x + y + 3z^2. The problem is symmetric in x, y, and z, so adding or subtracting the gradient equations can help find the other critical points.
  • #1
gassi
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Homework Statement



Find and classify the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3

Homework Equations


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The Attempt at a Solution



df/dx = y + z +3x^2, df/dy = x + z + 3y^2 and df/dz = x + y + 3z^2

a point x is a critacal point if the gradient equals 0.

Obviously (0,0,0) is a critical point but I´m not sure how to find the others.
 
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  • #2
gassi said:
Obviously (0,0,0) is a critical point but I´m not sure how to find the others.

Hi gassi! Welcome to PF! :smile:

Hint: the whole problem is completely symmetric in x y and z, isn't it?

So try adding or subtracting df/dx df/dy and df/dz. :smile:
 

Related to Classifying Critical Points of Multivariable Functions

What is the definition of a critical point?

A critical point is a point on a graph where the derivative is equal to zero or does not exist. It is also known as a stationary point.

How do I find critical points on a graph?

To find critical points on a graph, you must first find the derivative of the function. Then, set the derivative equal to zero and solve for the variable. The resulting values are the critical points.

Why are critical points important in mathematical analysis?

Critical points are important because they represent the maximum, minimum, or inflection points on a graph. They help us understand the behavior of a function and can be used to find the optimal values for a given problem.

What is the difference between a relative and absolute critical point?

A relative critical point is a point where the derivative is equal to zero, while an absolute critical point is a point where the derivative does not exist. Relative critical points can be local maxima or minima, while absolute critical points are global maxima or minima.

How do I determine the nature of a critical point?

To determine the nature of a critical point, you can use the second derivative test. If the second derivative is positive at the critical point, it is a local minimum. If the second derivative is negative, it is a local maximum. If the second derivative is zero, the test is inconclusive and you may need to use other methods to determine the nature of the critical point.

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