Term
absolute (global) maximum 

Definition
function value f(c) ∋
f(c) ≤ f(x) ∀x ∈ D_{f} 


Term
absolute (global) minimum 

Definition
function value f(c) ∋
f(c) ≤ f(x) ∀x ∈ D_{f} 


Term

Definition
If f is continuous on a closed interval, the f attains both an absolute maximum & minimum on that interval. 


Term

Definition
function value f(c) ∋
f(c) ≥ f(x) ∀x
in some open interval containing c, an interior point of D_{f} 


Term

Definition
function value f(c) ∋
f(c) ≤ f(x) ∀x
in some open interval containing c, an interior point of D_{f} 


Term
First Derivative Theorem for Local Extreme Values 

Definition
if f
in some open interval containing c has a local maximum/minimum value at an interior point c of its domain, and if f ' is defined at c, then:
f'(c) = 0 


Term

Definition
an interior point of the domain of f where f ' is 0 or undefined. 


Term
increasing on an interval 

Definition
if ∀x_{1}, x_{2} in the interval with x_{1} < x_{2} and f(x_{1}) < f(x_{2}) 


Term
decreasing on an interval 

Definition
if ∀x_{1}, x_{2} in the interval with x_{1} < x_{2} and f(x_{1}) > f(x_{2}). 


Term

Definition
if the function is increasing or decreasing on that interval 


Term

Definition
Suppose that c is a critical point of a continuous function of f, and that f is differentiable at every point in some interval containing c, except possibly at c itself.
1) if f' changes from  to + at c, then f has a local minimum at c.
2) if f' changes from + to  at c, then f has a local maximum at c. 


Term
Intermediate Value Theorem 

Definition
A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). 


Term

Definition
Suppose that y = f(x) is continous at every point of 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 ∈ (a,b) at which f'(c) = 0. 


Term

Definition
Suppose y = f(x) is continuous on a closed interval [a,b] and differentiable on the interval's interior (a,b). Then there is at least one point c ∈ (a,b) at which
f(b)  f(a) = f'(c)
b  a 


Term
concave up on an interval I if 

Definition


Term
concave down on an interval I if 

Definition


Term
Second Derivative Test for Concavity 

Definition
Suppose that f" exists on an interval I.
1) if f"(x) > 0 on I, then f is concave up on I.
2) if f"(x) < 0 on I, then f is concave down on I. 


Term

Definition
point on the domain where the function changes concavity. 


Term
Second Derivative Test for Local Extrema 

Definition
Suppose f" is continuous on an open interval that contains x = c.
1) if f'(c) = 0 and f"(c) < 0, then f has a local maximum at x = c.
2) if f'(c) = 0 and f"(c) > 0, then f has a local minimum at x = c.
3) if f'(c) = 0, and f"(c) = 0, then the test fails. 

