Question

For each problem, locate the critical points and evaluate the Hessian matrix at each critical point. a. \(f(x, y)=(x+y)^{2}\) b. \(f(x, y)=x^{2} y+x y^{2}\) c. \(f(x, y)=x^{4} y+x y^{4}-x y\) d. \(f(x, y, z)=x y+x z+y z\). e. \(f(x, y, z)=x^{2}+y^{2}+x z+2 z^{2}\)

Short answer

For a problem of this nature, the detailed results will vary based on each function. Therefore, it's impossible to provide a 'short answer' without the workings of each part of the problem.

Step-by-step solution

Calculating the gradient of the function

For each of \(f(x, y)\) or \(f(x, y, z)\), the gradient is calculated by finding the first derivative of the function with respect to x, y (and z if applicable). It's written as \(\nabla f = (f_x, f_y)\) or \(\nabla f = (f_x, f_y, f_z)\) for 2-dimensional and 3-dimensional functions respectively.

Solving for the critical points

The critical points are found by setting the components of the gradient to zero and solving the resulting equations for x, y (and z if applicable).

Finding the Hessian matrix

The Hessian matrix, \(H_f\), is a square matrix comprised of the second partial derivatives of the function. For a 2-dimensional function, the Hessian matrix is given by \(H_f = [[f_{xx}, f_{xy}], [f_{yx}, f_{yy}]]\), and for a 3-dimensional function it's \(H_f = [[f_{xx}, f_{xy}, f_{xz}], [f_{yx}, f_{yy}, f_{yz}], [f_{zx}, f_{zy}, f_{zz}]]\).

Evaluating the Hessian matrix at the critical points

Substitute the critical points into Hessian matrix. This step will help determine whether each critical point is a maximum, minimum, or a saddle point.

Key concepts

Gradient of a Function

In calculus, the gradient of a function is an essential tool to navigate the topography of multivariable functions. Imagine hiking on a hilly terrain and using a compass that points towards the steepest ascent direction — that's what the gradient does in the mathematical landscape.

The gradient of a function is a vector that contains all the partial derivatives of that function, and it represents the direction of the greatest rate of increase of the function. In more formal terms, for a function with two variables, the gradient is denoted as abla f = (f_x, f_y), where f_x and f_y are the first partial derivatives of the function with respect to x and y, respectively. In the case of functions with three variables, we also consider f_z, expanding our gradient vector to abla f = (f_x, f_y, f_z).

Finding the Gradient

To calculate the gradient of a function, take the partial derivative of the function with respect to each variable. This process is much like finding the slope in single-variable calculus, but now you have to do it for each dimension you're working with. Once calculated, the components of the gradient can be set to zero to locate the critical points of the function, which are points where the function doesn't increase or decrease in any direction.

Second Partial Derivatives

Once armed with the knowledge of the gradient, stepping into the realm of the second partial derivatives adds depth to our understanding. These derivatives provide information about the curvature of the function's graph. To grasp this concept, think of it as examining the bend of the path on your hike — is it curving upward towards a hill, or downward into a valley?

The second partial derivatives of a function are computed by taking the derivative of the first partial derivatives. For a function f(x, y), these would be represented as f_xx (the second partial derivative with respect to x), f_yy (the second partial derivative with respect to y), and the mixed derivatives f_xy and f_yx. It's worth noting that for functions which are 'well-behaved' (technically, those which are twice continuously differentiable), the mixed derivatives are equal, that is, f_xy = f_yx.

Constructing the Hessian Matrix

The collection of second partial derivatives can be elegantly organized into the Hessian matrix. For example, a two-variable function yields a 2x2 matrix: H_f = [[f_{xx}, f_{xy}], [f_{yx}, f_{yy}]]. This matrix serves as a valuable tool in determining the concavity at the critical points and is a cornerstone in classifying them as either relative maxima, minima, or saddle points.

Saddle Point Determination

Identifying the nature of a critical point can be like solving a mystery, where a saddle point is an intriguing twist in the narrative. A saddle point is neither a local maximum nor a local minimum; instead, it is a point where the function changes concavity. Picture a mountain pass or saddle — there's a rise in one direction and a fall in the other.

Saddle points can be determined by evaluating the Hessian matrix at the critical points. In two dimensions, one common test involves the determinant of the Hessian matrix. If the determinant is positive, the critical point might be a max or min, but if it's negative, the point is a saddle point. For instance, if we examined a critical point of a two-variable function and found that det(H_f) < 0, we could confidently state the presence of a saddle point at that location.

Interpretation of the Hessian Matrix

More in-depth insight can be gained by looking at the eigenvalues of the Hessian matrix. If all the eigenvalues are positive, the function is locally convex and the critical point is a minimum. If all are negative, it's locally concave and the point is a maximum. However, if the eigenvalues are mixed (i.e., some positive and some negative), the critical point is a saddle point. Understanding these properties helps students to better visualize the multidimensional shapes of functions, which is an invaluable skill in fields like optimization, economics, and engineering.