Optimization

Question 1

What are local maxima and minima?

Answer 1

Points where f’(x) is 0 or undefined

Question 2

Which points are candidates for local extrema?

Answer 2

Critical points

Question 3

How do you find critical points?

Answer 3

Take the derivative of a function and find where it is equal to 0 or undefined

Question 4

What two tests determine whether a critical point is a local max/min?

Answer 4

1st Derivative Test and 2nd Derivative Test

Question 5

According to the 1st Derivative Test, when is x = a a local max?

Answer 5

When f’(x) swtiches from positive to negative at x = a
When f(x) goes from increasing to decreasing

Question 6

According to the 1st Derivative Test, when is x = a a local min?

Answer 6

When f’(x) swtiches from negative to psotive at x = a
When f(x) goes from decreasing to increasing

Question 7

2nd Derivative Test: When f”(a) is negative, is x = a a local max or a local min?

Answer 7

Local max

Question 8

2nd Derivative Test: When f”(a) is positive, is x = a a local max or a local min?

Answer 8

Local min

Question 9

2nd Derivative Test: When f”(x) = 0, is x = a a local max or a local min?

Answer 9

Test inconclusive

Question 10

What is the absolute max of a function?

Answer 10

The x-coordinate on [a, b] where f(x) is as big as possible

Question 11

What is the absolute min of a function?

Answer 11

The x-coordinate on [a, b] where f(x) is as small as possible

Question 12

If f(x) is a continuous function on [a, b], what are the possible values for the absolute extrema?

Answer 12

The endpoints (x = a, x = b) and the critical points

Question 13

What are three steps to find absolute extrema?

Answer 13

1. Find the derivative of the function
2. Set it equal to zero (and see where it’s undefined) to determine the critical points
3. Substitute endpoints of the interval and critical points into f(x). Highest = absolute max, lowest = absolute min