Quiz 2 Flashcards
Extreme Value Theorem
If f(x) is continuous on the closed interval [a,b], then f(x) has both an absolute max value f(c) AND an absolute min value f(d) at some numbers c and d in [a,b].
Critical points
inferior points of the domain where f’(x) = 0 or f’(x) = DNE.
x-values
Not an endpoint
Steps to find extreme values
- Find all critical points ( set f’=0 and find where f’=DNE)
- Evaluate f(x) at the critical points
- Evaluate the endpoints
- Take the largest and smallest values (y values)
Extreme values occur…
where f’(x)=0, where f’(x)=DNE – cusp, corner
endpoints of a function’s domain
If f(x) is continuous on [a,b] and differentiable on (a,b), then,
- If f’(x) > 0 for all x in (a,b), then f(x) is increasing on [a,b]
- If f’(x)<0 for all x in (a,b), then f(x) is deceasing on [a,b]
At critical point c, if f’(x) changes from positive to negative …
f(x) has a local max at c
At critical point c, if f’(x) changes from negative to positive …
f(x) has a local min at c
At critical point c, if f’(x) does not change sign,
f(x) has no local extreme values at c
Concavity Test
A function is concave up when y’ is increasing…
f’‘(x) > 0
Concavity Test
A function is concave down when y’ is decreasing…
f”(x) < 0
Just because y”=0…
does not mean there is a point of inflection
Suppose f” is continuous near c…
If f’(c)=0 and f”(c)>0 then
f has a local minimum at c
Suppose f” is continuous near c…
If f’(c)=0 and f”(c)<0 then
f has a local maximum at c
Rolle’s Theorem
Suppose that y=f(x) is continuous at every point on the closed interval [a,b] and differentiable at every point of its interior (a,b). If f(a) = f(b), then there is at least one number c between a and b at which f’(c)=0.
Mean Value Theorem
If y=f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b), then there is at least one number c between a and b where
f’(c) = f(b) - f(a)
—————-
b - a
