91Ó°ÊÓ

In multivariable calculus, critical points are where the gradient (the vector of all partial derivatives) of a function equals zero. These points are crucial because they can indicate local maxima, minima, or saddle points, or they may be indeterminate.

To find critical points, we first compute the partial derivatives of the given function with respect to each variable. Next, we set these derivatives to zero and solve the resulting system of equations.

For example, consider the function \(f(x, y)=x^{3}-3 x y+y^{3}\). The partial derivatives are:

• \(f_x = 3x^{2} - 3y\)
• \(f_y = -3x + 3y^{2}\)

By setting \(f_x = 0\) and \(f_y = 0\), we find solutions \((0, 0)\) and \((1, 1)\). These are the critical points, which we will analyze further to determine their nature.

Partial derivatives reveal how a function changes as each individual variable changes, while keeping others constant. They are essential in multivariable calculus to understand the behavior of functions with several variables.

For a function \(f(x, y)\), a partial derivative with respect to \(x\), denoted \(f_x\), indicates how \(f(x, y)\) changes as \(x\) varies, with \(y\) fixed. Similarly, \(f_y\) is the derivative with respect to \(y\).

• Calculating partial derivatives involves differentiating as you normally would with one variable, treating all other variables as constants.
  For \(f(x, y) = x^3 - 3xy + y^3\), we found:
  
  • \(f_x = 3x^{2} - 3y\)
  • \(f_y = -3x + 3y^{2}\)

Understanding how to compute and interpret partial derivatives is a fundamental skill for analyzing multivariable functions.

The second partials test is a method used to classify critical points for functions of two variables. It helps to determine whether a critical point is a local minimum, local maximum, or a saddle point.

To perform this test, you first need the second order partial derivatives:

• \(f_{xx} \), the second derivative with respect to \(x\)
• \(f_{yy} \), the second derivative with respect to \(y\)
• \(f_{xy} = f_{yx} \), the mixed partial derivatives

The key to this test is calculating the Hessian determinant \(D(x, y) = f_{xx}f_{yy} - (f_{xy})^2\).

Next, evaluate \(D(x, y)\) at each critical point:

• If \(D > 0\) and \(f_{xx} > 0\), the point is a local minimum.
• If \(D > 0\) and \(f_{xx} < 0\), the point is a local maximum.
• If \(D < 0\), it is a saddle point.
• If \(D = 0\), the test is inconclusive.

For the critical points \((0, 0)\) and \((1, 1)\) in our example, the test showed \((0,0)\) is indeterminate, and \((1,1)\) is a local minimum. This method is invaluable for classifying points in multivariable functions.