Optimising functions of 2 variables

Question 1

In notes 8 when we looked at optimising functions of one variable what did we do ?

Answer 1

we used differentation to optimise a function of one variable.

Question 2

What did the first derative and second derative tell us when using differentation to optimise a function of one variable?

Answer 2

1st differential identified the stationary points

the second differential told us whether we had a max or min,

Question 3

When did we have a local maximum and when did we have a local minimum, when using second derative test to optimise a function of one variable?

Answer 3

local maximum at x= a if f’(a) = 0 and f’‘(a)<0

local minimum at x = a if f’(a) = 0 and f’‘(a)>0.

Question 4

In notes 11 our did we optimise functions of 2 variables and what was wrong with it?

Answer 4

The problem is that this only works for specific functions

Question 5

We are going to use a different method to now optimise functions of 2 variables, first of all what are we going to find and how do we find it?

Answer 5

We find partial deratives with respect to x and with respect to y and equate them meach to 0 and factorise or solve simultaneously to find the cricital points.

Question 6

Answer 6

Question 7

Answer 7

Question 8

When solving for critcial points and its not clear to complete the square or to factorise as in your partial deratives you have a mixture of x and y, how do we find the critical points?

Answer 8

We solve them simultaneously

Question 9

Answer 9

Whats important to remember is that when they equal 0, the first order partial deratives, they change sign when you have a mixture of x’s and y’s.

Question 10

Answer 10

Question 11

So after finding the critical points of a function of 2 variables what can we do?

Answer 11

We can determine their nature of the cricital points.

Question 12

How can we determine the nature of our crtical points of a function of 2 variables?

Answer 12

We have to find the hessian of the function

Question 13

So now with the hessian of the function what are the rules in determing whether we have a maximum or a minimum or a saddle point?

Answer 13

Question 14

Find the crictial points and determine its nature?

Answer 14

Question 15

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Question 16

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Question 17

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Question 18

How do we denote a firm maxmising revenue for selling 2 goods?

Answer 18

Question 19

What is the profit function of a firm, selling 2 goods x and y?

Answer 19

Question 20

Answer 20

TBA