Optimisation

Question 1

Define a maximum, minimum and a point of inflection, in terms of second derivatives of f(x).

Answer 1

At point (a, f(a)), the point can be defined as:
A maxima - if f’‘(a) < 0.
A minima - if f’‘(a) > 0.
A point of inflection - if f’‘(a) = 0 and the sign changes.

Question 2

What is the difference between a critical points and points of inflection?

Answer 2

Critical points are any type of stationary point on a graph, i.e. where f’(x) = 0.

A point of inflection is any point which f’(x) = 0 and the sign of f’‘(x) changes.

Question 3

What expansion can be used to explore the nature of a point of inflection (a, f(a))?

Answer 3

A Taylor series expansion.

Question 4

What a saddle point on a 3D graph?

Answer 4

One in which the point is a maximum in two axes but a minimum on the third.

Question 5

What type of point P can a hessian matrix be used for?

Answer 5

A hessian matrix acts upon a position vector P and its determinant gives the nature of point P, i.e. its type of critical point.

Question 6

What does a hessian matrix of function F look like?

Answer 6

[Fxx Fxy]
[Fxy Fyy]

Question 7

if detH > 0, what does this mean about position (a, b) of function F?
Similarly for detH < 0 and detH = 0.

Answer 7

if detH > 0, (a, b) is a local minimum or maximum of F. if Fxx > 0 or Fyy > 0, it is a minimum, and if Fxx < 0 or Fyy < 0, it is a maximum.

if detH < 0 then point (a, b) is a saddle point of F.

if detH = 0 further investigation is required.

Question 8

What does it mean to consider the lagrangian of function F(x, y)

Answer 8

The lagrangian is a function that must be used if F is subject to the constraints of some g(x, y). it looks like this:

F(x, y, λ) = f(x, y) + λg(x, y)

Consider it like a constant of integration which we must find when g(x, y) = 0 for a maxima or minima.