3.1-3.4 extrema reasoning Flashcards
What is the definition of extrema?
The min. and max. value of f on the interval in which f is defined (entire interval) (also called absolute extrema)
What is the Extreme Value Theorem?
If f is continuous on a closed interval [a,b], then f has a min. and max. on the interval.
What is the definition of Relative Extrema?
Point when f changes from inc. to dec. or vice versa/f’ changes from pos. to neg. or vice versa;
If f is defined at x=c, f(c) is a rel. max./min. if there exists an open interval containing c such that f(x)</>f(c) for all x in the interval
*Does not need to be continuous
What is the definition of a Critical Number?
If f is defined at c on an open interval, if f’(c) is 0 or undefined, then c is a critical number
What is the justification for absolute extrema?
The Candidate’s Test, where f is evaluated at each if its critical numbers on the open interval (include why they are CN) and f is evaluated at each endpoint of the interval; greatest val. is max., least val. is min.
What is Rolle’s Theorem?
If f is continuous on a closed interval and differentiable on the open interval and the value at each endpoint is equal, then there exists at least one c in the interval where f’(c)=0
*Max./min. NOT at endpoint
What is the justification for Rolle’s Theorem?
1) f is cont. on [a,b] and differentiable on (a,b)
2) f(a)=f(b), Rolle’s Thm. app.
What is the Mean Value Theorem?
If f is continuous on a closed interval and differentiable on the open interval, then there exists at least one c in the interval where f’(c)=[f(b)-f(a)]/(b-a)
(slope btw endpts = slope at some pt in the interval/exists a tan line parallel to sec line/avg rate of change=instantaneous rate of change)
What is the justification for MVT?
1) f is cont. on [a,b] and differentiable on (a,b), MVT app.
2) f’(c)=[f(b)-f(a)]/(b-a)
What is the justification for inc./dec. functions?
1) f inc on [a,b] bc f’(x)>0 [for all x on (a,b)
2) f dec on [a,b] bc f’(x)<0 [for all x on (a,b)
3) f constant on [a,b] bc f’(x)=0 [for all x on (a,b)]
What is the definition of the first derivative test?
If c is a critical no of f that is continuous on an open interval containing c and f is differentiable on the interval except possibly at c, f(c) is a rel. min/rel. max/neither
What is the justification for the First Derivative Test?
1) f’(c) is 0 or undefined at c
2) i. f(c) is a rel min bc f’(x) changes from neg to pos at c
ii. f(c) is a rel max bc f’(x) changes from pos to neg at c
iii. f(c) is neither bc f’(x) does not change signs
What is the definition of the test for concavity?
If f” exists on an open interval,
1) f cc up on (a,b) bc f”(x)>0
2) f cc down on (a,b) bc f”(x)<0
What is the definition of a Point of Inflection?
If (c, f(c)) is a POI on f, then f”(c) is 0 or undefined and f”(x) changes signs at c.
If the graph of a continuous f has a tan line at a point where its concavity changes from upward to downward (f”(x) from pos to neg) or vice versa, then that is a POI.
What is the justification for the second derivative test?
1) f’(c)=0 and f” exists on the open interval containing c
2) i. f(c) is a rel min if f”(c)>0
ii. f(c) is a rel max if f”(c)<0
* If f(c)=0, test fails and must use first derivative test
