4 Optimisation

Question 1

What is a global max

Answer 1

When the point is the greatest value of f(x) for all values of x

Question 2

What is a local max

Answer 2

When the point is the greatest value of f(x) within a given range

Question 3

If f’(x)=0 what can we say about this point?

Answer 3

It is an interior extreme point

Question 4

When can we be absolutely sure if a function has a global max and minimum?

Answer 4

If a continuous function exists in a closed and bounded interval then there is definitely a global max and min

Question 5

What is sufficient evidence that a point is an interior extreme point

Answer 5

The second order condition

Question 6

Sufficient first order condition test for global max

Answer 6

If f’(x)>=0 for x<=c and f’(x)<=0 for x>=c then x=c is a global max

Question 7

Sufficient first order condition for local max

Answer 7

If f’(x)>=0 for xc close to c then x=c is a local max

Question 8

What can be said if f’(x)=0 and f’’(x)<0?

Answer 8

Then x is a strict local max

Question 9

What can be said if f’(x)=0 and f’’(x)>0

Answer 9

Then x is a strict local min

Question 10

What can be said if f’(x)=0 and f’’(x)=0

Answer 10

Then no conclusion can be drawn

Question 11

If f’’(c)=0 and the f’’ changes sign at c what can be said about c?

Answer 11

It is an inflection point

Question 12

What is comparative statics used for?

Answer 12

To see how the behaviour of economic agents adjusts when economic conditions change

Question 13

General method for finding global min/max

Answer 13

1. Find critical points of f(x) for a<x<b
2. Evaluate the function at its critical points
3. Evaluate the function at x=a and x=b
4. Choose the highest/lowest of f(x) from 2) and 3)

Question 14

How do you find the global min/max for an interval that doesn’t include endpoints when the highest value of the function occurs at the endpoint

Answer 14

You can’t find the global min/max since it technically doesn’t exist