What are the critical points of a function with multiple variables?

In summary: So 2y(x+1)=0, so y=0 or x=-1In summary, to find the critical points of f(x,y) = 2x^3+xy^2+5x^2+y^2+100, you need to set the partial derivatives with respect to x and y equal to zero. This results in two equations, 6x^2+y^2+10x=0 and 2xy+2y=0. Solving these equations simultaneously gives the critical points (x,y) = (-1,0) or (0,0).
  • #1
Chadlee88
41
0

Homework Statement


Find all the critical points of f(x,y) = 2x^3+xy^2+5x^2+y^2+100

Homework Equations





The Attempt at a Solution



I'm really not sure how to do this question due the the x^3 term in the function. Could someone please advise how to start this.

Thanx :D
 
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  • #2
The critical points are where the partial derivatives of f with respect to x and y vanish simultaneously. Why should the x^3 be a problem?
 
  • #3
Partial derivatives

(6x^2+y^2+10x)i + (2xy+2y)j

how am i supposed to find values of x and y that make it equal to zero??

the partial derivative with respect to x is a quadratic function with both x and y terms. so I'm stuck!
 
  • #4
Chadlee88 said:
Partial derivatives

(6x^2+y^2+10x)i + (2xy+2y)j

how am i supposed to find values of x and y that make it equal to zero??

the partial derivative with respect to x is a quadratic function with both x and y terms. so I'm stuck!

You have two components and you can set each to zero. Then you have two equations and two unknowns.
 
  • #5
You shouldn't really have components anyway. The partial derivatives aren't a vector. Start with the second one 2xy+2y=0.
 
  • #6
Arent critical points those where either first derivative is 0 or not defined ?
 
  • #7
Dick said:
You shouldn't really have components anyway. The partial derivatives aren't a vector. Start with the second one 2xy+2y=0.

good point.
 

FAQ: What are the critical points of a function with multiple variables?

What are critical points of a function?

Critical points of a function are points on the graph of a function where the derivative is equal to zero or does not exist. These points are important because they can indicate where a function reaches its maximum or minimum value.

How do you find critical points of a function?

To find the critical points of a function, you must first take the derivative of the function. Then, set the derivative equal to zero and solve for the variable. The resulting value(s) will be the critical point(s) of the function.

What is the significance of critical points in calculus?

Critical points are significant in calculus because they can help determine the maximum or minimum value of a function. They also play a role in determining the concavity of a function and identifying points of inflection.

Can a function have more than one critical point?

Yes, a function can have multiple critical points. This can occur when the derivative of the function is equal to zero at more than one point or when the derivative does not exist at certain points.

How do critical points relate to optimization problems?

Critical points are essential in optimization problems because they indicate where a function reaches its maximum or minimum value. By finding the critical points and evaluating the function at those points, you can determine the optimal solution to the problem.

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