Last Updated
：2025/11/29
Rolle's theorem
Proper noun
(calculus)
The
theorem
that
any
real-valued
differentiable
function
that
attains
equal
values
at
two
distinct
points
must
have
a
point
somewhere
between
them
where
the
first
derivative
(the
slope
of
the
tangent
line
to
the
graph
of
the
function)
is
zero.
In
mathematical
terms,
if
f:ℝ→ℝ
is
differentiable
on
(a,b)
and
f(a)=f(b)
then
∃c∈(a,b):f'(c)=0.
Japanese Meaning
実数値の関数が区間の両端で同一の値をとるとき、その区間内のどこかに接線の傾き（微分係数）がゼロになる点が存在するという定理。 / 数学的には、区間(a, b)で微分可能な関数fに対し、f(a)=f(b)ならば、(a, b)内にある少なくとも一つの点cでf'(c)=0となることを保証する定理である。
Sense（1）
(calculus)
The
theorem
that
any
real-valued
differentiable
function
that
attains
equal
values
at
two
distinct
points
must
have
a
point
somewhere
between
them
where
the
first
derivative
(the
slope
of
the
tangent
line
to
the
graph
of
the
function)
is
zero.
In
mathematical
terms,
if
f:ℝ→ℝ
is
differentiable
on
(a,b)
and
f(a)=f(b)
then
∃c∈(a,b):f'(c)=0.
Quizzes for review
(calculus) The theorem that any real-valued differentiable function that attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero. In mathematical terms, if f:ℝ→ℝ is differentiable on (a,b) and f(a)=f(b) then ∃c∈(a,b):f'(c)=0.
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See correct answer
Rolle's theorem
In the lecture, we used Rolle's theorem to show there must be a point between a and b where the derivative of f is zero.
See correct answer
In the lecture, we used Rolle's theorem to show there must be a point between a and b where the derivative of f is zero.
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