Unit Three: Maxs and Mins Flashcards
Definition of extrema
Let f be defined on an interval I containing c
1) F(c) is a minimum of f in I if f(c) is less than or equal to f(x) for all x in I
2) F(c) is a maximum of f in I if f(c) is greater than or equal to f(x) for all x in I
The extreme value theroem
If f is continuous on [a,b] then f has both a max and a min on that interval (they can be the same number)
Critical numbers
Let f be defined at some x-value c. If f’(c) is zero, or if f is not differentiable at c, then c is a critical number
(Can help you find relative/absolute max’s and mins)
Rolle’s Theorem (MEMORIZE)
Let f be continuous on [a,b] and differentiable on the open interval (a,b). If f(a) equals f(b), then there is at least one number c in that open interval (a,b) such that f’(c) equals zero
The Mean Value Theorem (MEMORIZE)
If f in continuous on [a,b] and differentiable on (a,b), then there exists a c in (a,b) such that f’(c) equals (f(b)-f(a))/(b-a)
(Or the average slope)
How do you find the extrema on an interval?
Plug in the end points, then the critical numbers, use the biggest and the smallest as the max/min
How do you know when f is increasing, decreasing, or constant on an interval [a,b]?
If f’(x) is greater than zero (For all x in (a,b)), it is increasing.
If f’(x) is less than zero (For all x in (a,b)), it is decreasing.
If f’(x) is zero (For all x in (a,b)), it is constant.
How can you find the relative max’s and min’s for f?
If f’(x) changes from pos to neg at point c, (c,f(c)) is a relative max
If f’(x) changes from neg to pos at point c, (c,f(c)) is a relative min
If f’(x) changes from pos to pos or neg to neg at point c, (c,f(c)) is not a relative max or min
What is the meaning of concave up?
F is decreasing, then increasing
What is the meaning of concave down?
F is increasing, then decreasing
How do you determine if the graph of f is concave up or down?
It is up if f’ is increasing (f’’ is greater than zero), it is down if f’ is decreasing (f’’ is less than zero)
What is an inflection point?
If (c, f(c)) is a point of inflection of f, then either f’‘(c) is equal to zero or it does not exist at c. An inflection point is a “point” where f changes concavity.
What is the second derivative test? (will not have to use if you don’t want to)
Let f be a function such that f’(c) equals zero, and the second derivative of f exists on an interval containing v.
If f’‘(c) is greater than zero, f has a relative min at (c,f(c)).
If f’‘(c) is less than zero, f has a relative max at (c,f(c)).
If f’‘(c) is zero, f has neither.
How do you find y-intercepts?
X=0
How do you find x-intercepts?
Y=0, only pay attention to numerator