3.1-3.4 Theorems

Question 1

Extreme ValueTheorem

IF f is _______ on open/closed interval (a,b)/[a,b], THEN f….on the interval

Answer 1

Extreme Value Theorem
continuous on closed interval [a,b]
f has both a minimum and maximum

Question 2

Rolle’s Theorem
Let f be _____ on open/closed interval (a,b)/[a,b]
and _____ on open/closed interval (a,b)/[a,b].
If _____ THEN there is at least one number c in (a,b) such that….

Answer 2

Rolle's Theorem
continuous on closed interval [a,b]
differentiable on open interval (a,b)
IF f(a)=f(b)
such that f'(c)=0

Question 3

Mean Value Theorem
IF f is _______ on open/closed interval (a,b)/[a,b]
and _____ on open/closed interval (a,b)/[a,b]
THEN there exists a number c in (a,b) such that….

Answer 3

Mean Value Theorem
continuous on closed interval [a,b]
differentiable on open interval (a,b)
f'(c)= [f(b)-f(a)]/[b-a]......
(f'(c)=slope of line connecting endpts. (secant))

Question 4

The___ Derivative Test
Let c be a ____ of a function f
that is _______ on open/closed interval containing c.
IF f is _____ on the interval, ……., THEN:
1. If f’(x)…. at c, then f has….. at ___
2.If f’(x)…. at c, then f has….. at ___
3.If f’(x)…. at c, then…..

Answer 4

The First Derivative Test
c is a critical number
continuous on open interval
differentiable, except possibly at c
1. the sign changes from - to +,… relative min…(c,f(c))
2. the sign changes from + to -,… relative max…(c,f(c))
3. the sign doesn’t change… f(c) is neither a relative max or min

Question 5

The\_\_\_ Derivative Test
Let f be a function such that \_\_\_\_ and ..... on an open/closed interval containing c.
1. If f''(c)...., then....
2.If f''(c)...., then....
3.If f''(c)...., then....
What should you do if #3 occurs?

Answer 5

The Second Derivative Test
f’(c)=0 and the second derivative of f exists… open
1. f’‘(c) > 0, then f(c) is a relative min
2.f’‘(c) < 0, then f(c) is a relative max
3. f’‘(c) = 0, then the test fails. (Use the First Derivative Test)