# Week 3 Flashcards

How to determine local extrama?

- 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

What is the sign change test?

- Plug in values above and below stationary point
- Let a < b

if f’(a)

Difference between local and global extrema?

local - compare values near point a f(a)

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

Do global extrema ALWAYS exist?

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

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

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.

What is a continous function?

find out

What does f’(x) tell us?

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).

What does f’‘(x) tell us?

This tells us the

What does f’‘(x) tell us?

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.

Where can possible extrema be found?

let f: [a, b] –> R

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

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

If f’‘(x) >= 0 then convex

If f’‘(x) <= 0 then concave

What does convexity/concavity tell us about extrema?

If convex function, local minima is global minima

If concave, local maxima is global maxima