**Definition of a concave up curve:** f(x) is "concave up"
at x_{0} if and only if f '(x) is increasing at x_{0}
**Definition of a concave down curve:** f(x) is "concave down"
at x_{0} if and only if f '(x) is decreasing at
x_{0}

**The second derivative test:** If f ''(x) exists
at x_{0} and is positive, then f ''(x) is concave
up at x_{0}. If f ''(x_{0}) exists and
is negative, then f(x) is concave down at x_{0}. If f
''(x) does not exist or is zero, then the test fails.

**Local (Relative) Extrema**
**Definition of a local maxima:** A function f(x) has a local maximum
at x_{0} if and only if there exists some interval I containing
x_{0} such that f(x_{0}) >= f(x) for all x in I.

**Definition of a local minima:** A function f(x) has a local minimum
at x_{0} if and only if there exists some interval I containing
x_{0} such that f(x_{0}) <= f(x) for all x in I.

**Occurrence of local extrema:** All local extrema occur at critical
points, but not all critical points occur at local extrema.

**The first derivative test for local extrema:** If f(x) is increasing
(f '(x) > 0) for all x in some interval (a, x_{0}]
and f(x) is decreasing (f '(x) < 0) for all x in some
interval [x_{0}, b), then f(x) has a local maximum at x_{0}.
If f(x) is decreasing (f '(x) < 0) for all x in some
interval (a, x_{0}] and f(x) is increasing (f '(x)
> 0) for all x in some interval [x_{0}, b), then f(x) has
a local minimum at x_{0}.

**The second derivative test for local extrema:** If f '(x_{0})
= 0 and f ''(x_{0}) > 0, then f(x) has a local
minimum at x_{0}. If f '(x_{0}) = 0 and
f ''(x_{0}) < 0, then f(x) has a local maximum
at x_{0}.

**Absolute Extrema**

**Definition of absolute maxima:** y_{0} is the "absolute
maximum" of f(x) on I if and only if y_{0} >= f(x) for all
x on I.

**Definition of absolute minima:** y_{0} is the "absolute
minimum" of f(x) on I if and only if y_{0} <= f(x) for all
x on I.

**The extreme value theorem:** If f(x) is continuous in a closed
interval I, then f(x) has at least one absolute maximum and one absolute
minimum in I.

**Occurrence of absolute maxima:** If f(x) is continuous in a closed
interval I, then the absolute maximum of f(x) in I is the maximum value
of f(x) on all local maxima and endpoints on I.

**Occurrence of absolute minima:** If f(x) is continuous in a closed
interval I, then the absolute minimum of f(x) in I is the minimum value
of f(x) on all local minima and endpoints on I.

**Alternate method of finding extrema:** If f(x) is continuous
in a closed interval I, then the absolute extrema of f(x) in I occur
at the critical points and/or at the endpoints of I.

*(This is a less specific form of the above.)*

**Definition of an increasing function:** A function f(x)
is "increasing" at a point x_{0} if and only if there exists some
interval I containing x_{0} such that f(x_{0}) > f(x)
for all x in I to the left of x_{0} and f(x_{0}) <
f(x) for all x in I to the right of x_{0}.
**Definition of a decreasing function:** A function f(x) is "decreasing"
at a point x_{0} if and only if there exists some interval I
containing x_{0} such that f(x_{0}) < f(x) for all
x in I to the left of x_{0} and f(x_{0}) > f(x) for
all x in I to the right of x_{0}.

**The first derivative test:** If f '(x_{0})
exists and is positive, then f '(x) is increasing at x_{0}.
If f '(x) exists and is negative, then f(x) is decreasing
at x_{0}. If f '(x_{0}) does not exist
or is zero, then the test tells fails.