Avon High School
AP Calculus AB
UNIT 4 REVIEW
Theorems, Tests, Etc.
Extreme Value Theorem
1.) f is continuous on a closed interval  a, b
Conclusion: f has both an absolute maximum and absolute minimum on that interval
Rolle’s Theorem
1.) f is continuous on the closed interval  a, b
2.) f is differentiable on the open interval  a, b 
3.) f  a   f  b 
Conclusion: there is at least one number c in  a, b  such that f   c   0
Mean Value Theorem
1.) f is continuous on the closed interval  a, b
2.) f is differentiable on the open interval  a, b 
Conclusion: there exists a number c in  a, b  such that f   c  
f b   f  a 
ba
Test for Increasing/Decreasing
1.) find critical numbers by f   c   0 and f   c  DNE
2.) test intervals on each side of critical numbers
Conclusion: if f   x   0 for all x in the interval, f  x  is increasing in the interval;
if f   x   0 for all x in the interval, f  x  is decreasing in the interval;
if f   x   0 for all x in there interval, f  x  is constant in the interval
First Derivative Test (Relative Extrema)
1.) find critical numbers by f   c   0 and f   c  DNE
2.) test intervals on each side of critical numbers
Conclusion: if f   x  changes from negative to positive at c, then f has a relative minimum at c, f  c 



if f   x  changes from positive to negative at c, then f has a relative maximum at c, f  c 
note: c must be in the domain of f  x 

Avon High School
AP Calculus AB
UNIT 4 REVIEW
Theorems, Tests, Etc.
Test for Concavity
1.) f   x  exists on an open interval I
2.) locate x-values at which f   x   0 or f   x  DNE
Conclusion: if f   x   0 for all x in I, then the graph of f is concave upward on I;
if f   x   0 for all x in I, then the graph of f is concave downward on I
Points of Inflection
1.) f   c   0 or f   c  DNE
2.) f   x  changes from negative to positive or from positive to negative at c


Conclusion: then c, f  c  is a point of inflection
Second Derivative Test (Relative Extrema)
1.) find critical numbers by f   c   0 and f   c  DNE
2.) f   x  exists on an open interval containing c


f   c   0 , then f has a relative maximum at  c, f  c  
Conclusion: if f   c   0 , then f has a relative minimum at c, f  c 
if