extreme values

MAXIMUM AND MINIMUM VALUES

D = domain, c = point in domain

absolute maximum - f(c) ≥ f(x) for all x in D

absolute minimum - f(c) ≤ f(x) for all x in D

local maximum - f(c) ≥ f(x) when x is near c

local minimum - f(c) ≤ f(x) when x is near c

local extreme values cannot occur at endpoints

FERMAT’S THEOREM

if f(c) is a local extreme value, then f’(c) = 0

-the converse of the theorem is false

-there can still be extreme values even if f’(c) does not exist

CLOSED INTERVAL METHOD

how to find absolute extreme values of f in [a,b]:

1. find the values of f at the critical numbers of f in (a,b)
2. find the values of f at the endpoints (first and last point of the interval)
3. highest value of f is abs max, lowest is abs min

HOW TO FIND CRITICAL NUMBERS

critical numbers - numbers where f’(x) = 0 or DNE

1. differentiate f(x)
2. set f’(x) equal to zero
3. solve for x

FOR THE LOVE OF ALL THAT IS HOLY, ANSWER THESE

if you’re done…