List of basic formulas and conditions for finding extrema of functions of two and three variables.
Extrema of functions of two variables
- Function of two variables: ( f(x, y) )
- 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
]
- 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}
]
- Discriminant:
- Calculate the discriminant:
[
D = f_{xx} f_{yy} – (f_{xy})^2
]
- 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
- Function of three variables: ( f(x, y, z) )
- 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
]
- Second derivatives:
- Find the second derivatives:
[
f_{xx}, \quad f_{yy}, \quad f_{zz}, \quad f_{xy}, \quad f_{xz}, \quad f_{yz}
]
- 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}
]
- 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.