exists_hasDerivAt_eq_zero
theorem exists_hasDerivAt_eq_zero :
∀ {f f' : ℝ → ℝ} {a b : ℝ},
a < b →
ContinuousOn f (Set.Icc a b) →
f a = f b → (∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) → ∃ c ∈ Set.Ioo a b, f' c = 0
**Rolle's Theorem (HasDerivAt version)**: For any real valued functions `f` and `f'`, and any two real numbers `a` and `b` such that `a < b`, if the function `f` is continuous on the closed interval from `a` to `b`, and `f(a) = f(b)`, and at all points `x` in the open interval from `a` to `b`, `f` has `f'(x)` as its derivative at point `x`, then there exists a point `c` in the open interval from `a` to `b` such that the derivative of `f` at `c` is zero. In mathematical terms, for a continuous function `f` on `[a, b]` with `f(a) = f(b)`, there exists `c` in `(a, b)` such that `f'(c) = 0`.
More concisely: If a real-valued function `f` is continuous on a closed interval `[a, b]` where `a < b` and `f(a) = f(b)`, then there exists a point `c` in the open interval `(a, b)` such that `f'(c) = 0`.
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exists_hasDerivAt_eq_zero'
theorem exists_hasDerivAt_eq_zero' :
∀ {f f' : ℝ → ℝ} {a b l : ℝ},
a < b →
Filter.Tendsto f (nhdsWithin a (Set.Ioi a)) (nhds l) →
Filter.Tendsto f (nhdsWithin b (Set.Iio b)) (nhds l) →
(∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) → ∃ c ∈ Set.Ioo a b, f' c = 0
This is a version of **Rolle's Theorem** for a function defined on an open interval. The theorem states that if a real-valued function `f` has a derivative `f'` on the open interval `(a, b)` and this function `f` has the same limit `l` when we approach `a` from the right (denoted as `𝓝[>] a`) and `b` from the left (denoted as `𝓝[<] b`), then there exists a point `c` within the open interval `(a, b)` such that the derivative of `f` at `c` is zero, i.e., `f' c = 0`.
More concisely: If a real-valued function `f` with derivative `f'` on the open interval `(a, b)` has the same limit `l` as `a` is approached from the right and as `b` is approached from the left, then there exists a point `c` in `(a, b)` such that `f'(c) = 0`.
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exists_deriv_eq_zero
theorem exists_deriv_eq_zero :
∀ {f : ℝ → ℝ} {a b : ℝ}, a < b → ContinuousOn f (Set.Icc a b) → f a = f b → ∃ c ∈ Set.Ioo a b, deriv f c = 0 := by
sorry
**Rolle's Theorem** (deriv version): This theorem states that given a real-valued function `f`, and two real numbers `a` and `b` where `a < b`, if the function `f` is continuous on the closed interval from `a` to `b` (inclusive of `a` and `b`), and `f(a) = f(b)`, then there exists a real number `c` in the open interval from `a` to `b` (exclusive of `a` and `b`) such that the derivative of `f` at `c` is zero. In mathematical terms, this says that under these conditions, there's at least one point in the interval `(a,b)` where the function `f` has a horizontal tangent.
More concisely: If a real-valued function `f` is continuous on a closed interval `[a, b]` and `f(a) = f(b)`, then there exists a number `c` in the open interval `(a, b)` such that `f'(c) = 0`.
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exists_deriv_eq_zero'
theorem exists_deriv_eq_zero' :
∀ {f : ℝ → ℝ} {a b l : ℝ},
a < b →
Filter.Tendsto f (nhdsWithin a (Set.Ioi a)) (nhds l) →
Filter.Tendsto f (nhdsWithin b (Set.Iio b)) (nhds l) → ∃ c ∈ Set.Ioo a b, deriv f c = 0
The theorem `exists_deriv_eq_zero'` is a version of **Rolle's Theorem** for a function on an open interval. It states that for any real-valued function `f` and any real numbers `a`, `b`, and `l`, if `a` is strictly less than `b`, `f` tends to limit `l` within the interval greater than `a`, and `f` also tends to the same limit `l` within the interval less than `b`, then there exists a number `c` within the open interval between `a` and `b` such that the derivative of `f` at `c` is equal to zero. This version does not require the function `f` to be differentiable, as it defines the derivative of `f` at `c` to be zero whenever `f` is not differentiable at `c`.
More concisely: If real-valued function `f` has a limit `l` at both endpoints of an open interval `(a, b)` where `a < b`, then there exists a number `c` in the interval such that `f'(c) = 0` or `f` is not differentiable at `c`.
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