Question

Examine the function for relative extrema and saddle points. $$ f(x, y)=x^{3}-3 x y+y^{3} $$

Short answer

The function has a saddle point at (0,0), a local maximum at (-1,-1), and a local minimum at (1,1).

Step-by-step solution

Calculate first partial derivatives and find critical points

The partial derivative of \(f(x, y)\) with respect to \(x\) is \(3x^2 - 3y\), and with respect to \(y\) is \(-3x + 3y^2\). Find the critical points by setting these partial derivatives to zero and solving the resulting system of equations.

Calculate second partial derivatives and form the Hessian matrix

The second partial derivatives are \(f_{xx} = 6x, f_{yy} = 6y\), and \(f_{xy} = f_{yx} = -3\). Create a Hessian matrix from these second derivatives: \(H = [f_{xx}, f_{xy}; f_{yx}, f_{yy}]\).

Evaluate determinant of Hessian at critical points

The determinant of the Hessian matrix, \(D\), is \(f_{xx}f_{yy} - f_{xy}f_{yx}\). Evaluate \(D\) at the critical points found in the first step.

Classify critical points as relative extrema or saddle points

A critical point is a saddle point if \(D < 0\). If \(D > 0\), then it is a local minimum if \(f_{xx} > 0\), and a local maximum if \(f_{xx} < 0\).

Key concepts

Partial Derivatives

Understanding partial derivatives is crucial when analyzing the behavior of multivariable functions such as
\( f(x, y)=x^{3}-3xy+y^{3} \). Partial derivatives measure how a function changes as each variable changes, while the other variables are held constant. For instance, the partial derivative of \( f \) with respect to \( x \), denoted as \( f_x \), reflects the rate at which \( f \) changes with \( x \) when \( y \) is constant.

When calculating partial derivatives, each variable is treated independently. As seen in our exercise, the function's first partial derivatives are obtained by differentiating with respect to \( x \) and to \( y \) independently. It is essential to find these derivatives accurately as they are the building blocks for identifying critical points, which are candidates for being relative extrema or saddle points.

Critical Points

Critical points are where the function's partial derivatives are all zero or undefined. They are important because they are potential locations for local minima, local maxima, or saddle points.

In our example, the critical points of \( f \) are found by setting the first partial derivatives equal to zero and solving for \( x \) and \( y \). This yields a system of equations:

• \( 3x^2 - 3y = 0 \)
• \( -3x + 3y^2 = 0 \)

By solving this system, you obtain values for \( x \) and \( y \) which are the coordinates of the critical points. These are the locations we will further analyze to determine whether they are local maxima, local minima, or saddle points.

Hessian Matrix

The Hessian matrix is a square matrix of second-order partial derivatives of a function. It is a key tool in multivariable calculus, used specifically to analyze the curvature of surfaces and classify critical points.

In our function \( f \), the Hessian matrix at any point \( (x, y) \) is:

• \( H = \begin{bmatrix} 6x & -3 \ -3 & 6y \end{bmatrix} \)

The determinant of the Hessian matrix at a critical point, represented as \( D \), helps identify if that point is a relative minimum, relative maximum, or a saddle point. The values and signs of the determinant and the elements of the Hessian matrix give clues about the nature of the function around the critical points.

Local Minimum and Maximum

Local minima and maxima are types of relative extrema. A local maximum is a point where the function's value is greater than all nearby points, while a local minimum is where it's lower than all nearby points. These points are found among the critical points of a function.

Following our exercise's procedure, after finding the Hessian matrix at a critical point, we determine the type of extremum by looking at the determinant \( D \) and the sign of \( f_{xx} \). If \( D > 0 \) and \( f_{xx} > 0 \), the point is a local minimum, and if \( D > 0 \) and \( f_{xx} < 0 \), it is a local maximum. Otherwise, if \( D < 0 \), the point is not an extremum but a saddle point. These criteria are essential for understanding the topography of the function's graph around critical points.