Calculus BC: Applications of the Derivative
Terms
Critical Point
-
A number
x
in the domain of a function
f
such that
f'(x) = 0
.
Global Maximum
-
A value
f (x
0)
that is greater than or equal to any other value
f (x)
that
f
takes on
over its domain.
Global Minimum
-
A value
f (x
0)
that is less than or equal to any other value
f (x)
that
f
takes on over
its domain.
Inflection Point
-
A number
x
in the domain of a function
f
such that
f’’(x) = 0
.
Local Maximum
-
A value
f (x
0)
that is greater than or equal to any other value
f (x)
for
x
in some
interval about
x
0
.
Local Minimum
-
A value
f (x
0)
that is less than or equal to any other value
f (x)
for
x
in some
interval about
x
0
.
First Derivative Test
-
A critical point
x
0
of a function
f
is a local maximum if the first derivative
f'
changes sign from
positive to negative at
x
0
. Correspondingly,
x
0
is a local minimum is
f'
changes sign from
negative to positive there.
Second Derivative Test
-
A critical point
x
0
of a function
f
is a local maximum if the second derivative
f''(x
0)
is
negative. It is a local minimum if
f''(x
0)
is positive. (It is also possible that
f''(x
0) = 0
, in
which case the critical point is also an inflection point.)
Concave Up
-
A function
f (x)
is concave up at
x
0
if
f''(x
0) > 0
.
Concave Down
-
A function
f (x)
is concave down at
x
0
if
f''(x
0) < 0
.
