Week 3

Question 1

How to determine local extrama?

Answer 1

• Determine f’(x)
• f’(x) = 0 = stationary point (slope has no gradient / flat)
• Determine f’‘(x)
• If f’‘(x) > 0, local minima, if f’‘(x) < 0, local maxima
• If f’‘(x) = 0, inconclusive
• If inconclusive have to do sign change test

Question 2

What is the sign change test?

Answer 2

• Plug in values above and below stationary point
• Let a < b
  if f’(a)

Question 3

Difference between local and global extrema?

Answer 3

local - compare values near point a f(a)

global - compare ALL values to point a f(a)

Question 4

Do global extrema ALWAYS exist?

Answer 4

No, domain is something such as R then no global extrema as there can always be a lower or higher extrema.

Question 5

Can there be a global extrema AND a local extrema? Do they need to be the same?

Answer 5

Yes there are some situtations where there can be both.
i.e. In continous functions with closed intervals

No they do not always need to be the same.

Question 6

What is a continous function?

Answer 6

find out

Question 7

What does f’(x) tell us?

Answer 7

This tells us the slope of the tangent at a specific point a, f(a).

Tells us the rate of change of the function f(x).

Question 8

What does f’‘(x) tell us?

Answer 8

This tells us the

Question 8

What does f’‘(x) tell us?

Answer 8

This tells us the rate of change of f’(x).

If f’‘(x) > 0 then local minima, if f’‘(x) then local maxima, if f’‘(x) = 0 then inconclusive.

Question 9

Where can possible extrema be found?

let f: [a, b] –> R

Answer 9

• boundary points (a or b, f’(x) NOT present)
• stationary points (required, candidates)
• singular points (f’(x) not present)

Question 10

How to determine if f(x) is convex/concave?

Answer 10

If f’‘(x) >= 0 then convex

If f’‘(x) <= 0 then concave

Question 11

What does convexity/concavity tell us about extrema?

Answer 11

If convex function, local minima is global minima

If concave, local maxima is global maxima