What are the Critical Points of the Function ( f(x, y) xy^2 - x^2y^2x^4 )?

Understanding critical points in a multivariable function is essential in analysis and optimization. In this article, we will determine the critical points of the function ( f(x, y) xy^2 - x^2y^2x^4 ) by finding its partial derivatives and solving for the conditions under which they are zero or do not exist.

Introduction to Critical Points

A critical point of a function ( f(x, y) ) is a point where at least one of the partial derivatives, ( f_x(x, y) ) or ( f_y(x, y) ), is zero or does not exist. For the function ( f(x, y) xy^2 - x^2y^2x^4 ), we need to find the partial derivatives ( f_x(x, y) ) and ( f_y(x, y) ).

Calculating Partial Derivatives

First, let's calculate the partial derivative of ( f(x, y) ) with respect to ( x ):
[ f_x(x, y) frac{partial}{partial x} (xy^2 - x^2y^2x^4) ]

Using basic differentiation rules, we can derive that:
[ f_x(x, y) y^2 - 2xy^2 - 4x^3y^2x^4 ]

Simplifying, we get:
[ f_x(x, y) y^2 - 2xy^2 - 4x^7 ]

Next, we calculate the partial derivative with respect to ( y ):
[ f_y(x, y) frac{partial}{partial y} (xy^2 - x^2y^2x^4) ]

We derive that:
[ f_y(x, y) 2xy - 2x^2y ]

Setting Partial Derivatives to Zero

Now, we need to find the points where both partial derivatives are simultaneously zero:

Setting ( f_x(x, y) 0 ) and ( f_y(x, y) 0 ):
[ y^2 - 2xy^2 - 4x^7 0 ]
[ 2xy - 2x^2y 0 ]

Let's solve these equations step by step.

Solving ( f_y(x, y) 0 )

From ( 2xy - 2x^2y 0 ):

[ 2xy(1 - x) 0 ]

This equation gives us two possibilities:

( x 0 ) ( y 0 ) ( x 1 )

Solving ( f_x(x, y) 0 ) for ( x 0 )

When ( x 0 ):

[ y^2 0 ]

This leads to:

[ y 0 ]

Thus, one critical point is ( (0, 0) ).

Solving ( f_x(x, y) 0 ) and ( f_y(x, y) 0 ) for ( y 0 )

When ( y 0 ):

[ 2xy - 2x^2y 0 ] becomes ( 0 0 ), which is always true.

This also leads to the point ( (0, 0) ).

Solving ( f_x(x, y) 0 ) for ( x 1 )

When ( x 1 ):

[ y^2 - 2y^2 - 4 0 ]

Simplifying:

[ -y^2 - 4 0 ]

[ y^2 -4 ]

This equation has no real solutions, so we do not get any additional critical points from this case.

Summary of Critical Points

Combining all the cases, the critical points of the function ( f(x, y) xy^2 - x^2y^2x^4 ) are:

( (0, 0) ) ( (1, -2) ) ( (1, 2) )

Conclusion

By performing a detailed analysis of the function and its partial derivatives, we have identified the critical points. These points are essential for further analysis in optimization and understanding the behavior of the function.

References

[1] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.

Further Reading

Explore more on Mathematics Tutorial on Critical Points.