List of basic formulas and conditions for finding extrema of functions of two and three variables.

Extrema of functions of two variables

  1. Function of two variables: ( f(x, y) )
  2. Critical points:
  • Find the partial derivatives:
    [
    f_x = \frac{\partial f}{\partial x}, \quad f_y = \frac{\partial f}{\partial y}
    ]
  • Set up the equations:
    [
    f_x = 0, \quad f_y = 0
    ]
  1. Second derivatives:
  • Find the second derivatives:
    [
    f_{xx} = \frac{\partial^2 f}{\partial x^2}, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2}, \quad f_{xy} = \frac{\partial^2 f}{\partial x \partial y}
    ]
  1. Discriminant:
  • Calculate the discriminant:
    [
    D = f_{xx} f_{yy} – (f_{xy})^2
    ]
  1. Classification of critical points:
  • If ( D > 0 ) and ( f_{xx} > 0 ) — minimum.
  • If ( D > 0 ) and ( f_{xx} < 0 ) — maximum.
  • If ( D < 0 ) is a saddle point.
  • If (D = 0) – the test is not defined.

Extrema of functions of three variables

  1. Function of three variables: ( f(x, y, z) )
  2. Critical points:
  • Find the partial derivatives:
    [
    f_x = \frac{\partial f}{\partial x}, \quad f_y = \frac{\partial f}{\partial y}, \quad f_z = \frac{\partial f}{\partial z}
    ]
  • Set up the equations:
    [
    f_x = 0, \quad f_y = 0, \quad f_z = 0
    ]
  1. Second derivatives:
  • Find the second derivatives:
    [
    f_{xx}, \quad f_{yy}, \quad f_{zz}, \quad f_{xy}, \quad f_{xz}, \quad f_{yz}
    ]
  1. Gessian:
  • Составьте матрицу Гессе:
    [
    H = \begin{bmatrix}
    f_{xx} & f_{xy} & f_{xz} \
    f_{xy} & f_{yy} & f_{yz} \
    f_{xz} & f_{yz} & f_{zz}
    \end{bmatrix}
    ]
  1. Classification of critical points:
  • Determine the sign of the determinant of the Hessian matrix. If the determinant is positive and ( f_{xx} > 0 ) — minimum; if negative — maximum; if equal to zero — the test is undefined.

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