Higher order derivatives. Higher order differentials. Extrema of multivariable functions

1. Lecture 3

Higher order derivatives. Higher
order differentials. Extrema of
multivariable functions

2. Higher order derivatives

f ( x, y )
Mixed derivatives
f ( x, y )
x
2 f ( x, y )
x 2
Example 3.1
2 f ( x, y )
x y
f ( x, y )
y
2 f ( x, y )
y x
2 f ( x, y )
y 2

3.

4.

5.

6. Higher order derivatives for compositions of functions

7.

8.

9. Higher order differentials

10.

11. Differential of k-order

12.

13. Higher order differentials for composition of function

Differential for the case
of independent variables
Expressions for differentiating of
differentials dx1 and so on

14.

Differentials of higher orders for
multivariable function are not invariant.

15. Extrema of multivariable function. Necessary conditon.

16.

Necessary condition of extrema
or

17. Sufficient condition

18. Second partial derivative test