For all of the matches below,
x
=
c
\displaystyle {x}={c}
x
=
c
is a critical number of
f
\displaystyle {f}
f
.
-
a
b
c
If
f
′
(
x
)
\displaystyle {f}'{\left({x}\right)}
f
′
(
x
)
does not change sign at
x
=
c
\displaystyle {x}={c}
x
=
c
then
-
a
b
c
If
f
′
(
x
)
\displaystyle {f}'{\left({x}\right)}
f
′
(
x
)
changes from positive to negative at
x
=
c
\displaystyle {x}={c}
x
=
c
then
-
a
b
c
If
f
′
(
x
)
\displaystyle {f}'{\left({x}\right)}
f
′
(
x
)
changes from negative to positive at
x
=
c
\displaystyle {x}={c}
x
=
c
then
f
\displaystyle {f}
f
has a local minimum at
(
c
,
f
(
c
)
)
\displaystyle {\left({c},{f{{\left({c}\right)}}}\right)}
(
c
,
f
(
c
)
)
.
f
\displaystyle {f}
f
has a local maximum at
(
c
,
f
(
c
)
)
\displaystyle {\left({c},{f{{\left({c}\right)}}}\right)}
(
c
,
f
(
c
)
)
.
(
c
,
f
(
c
)
)
\displaystyle {\left({c},{f{{\left({c}\right)}}}\right)}
(
c
,
f
(
c
)
)
is neither a local maximum nor a local minimum.
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