5. Multivariate Calculus

Question 1

What is the partial derivative?

Answer 1

The derivative of a function with respect to one of its variables

Question 2

How many first order and second order derivatives will a function on n variables have?

Answer 2

N first order derivatives
N^2 second order derivatives

Question 3

What does a cross partial derivative show?

Answer 3

How fi changes as the value of the variable xj changes

Question 4

Youngs theorem

Answer 4

For functions with continuous first and second order partial derivatives fij =fji for all i,j and at all points

Question 5

How can we use the hessian matrix to determine if a function is convex/concave?

Answer 5

A function is convex/concave if and only if the associated Hessian is positive/negative semidefinite

Question 6

If a hessian matrix of x* is negative definite what can we say about the point x*?

Answer 6

(Strict) local max

Question 7

If a hessian matrix of x* is positive definite what can we say about the point x*?

Answer 7

(Strict) local min

Question 8

If a hessian matrix of x* is indefinite what can we say about the point x*?

Answer 8

It is a saddle point

Question 9

When do we check the bordered hessian condition?

Answer 9

To check if the solution we find for the Lagrange is really a max point

Question 10

What is the easiest way to use the Lagrange method to minimise problems

Answer 10

Use it the same way as normal but maximimse -f