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

[Chadlee88]

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

 

[Dick]

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?

 

[Chadlee88]

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!

 

[Pythagorean]

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.

 

[Dick]

You shouldn't really have components anyway. The partial derivatives aren't a vector. Start with the second one 2xy+2y=0.

 

[f(x)]

Arent critical points those where either first derivative is 0 or not defined ?

 

[Pythagorean]

You shouldn't really have components anyway. The partial derivatives aren't a vector. Start with the second one 2xy+2y=0.

good point.