Theorems · Theorem · real analysis
exists_deriv_eq_slope
- #75 of the 100 theorems: The Mean Value Theorem
- 1000+ list: Mean value theorem
∀ (f : ℝ → ℝ) {a b : ℝ},
a < b →
ContinuousOn f (Set.Icc a b) →
DifferentiableOn ℝ f (Set.Ioo a b) → ∃ c ∈ Set.Ioo a b, deriv f c = (f b - f a) / (b - a)Lagrange's Mean Value Theorem, deriv version.
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Set.Iccstatement and proof · cited by 1,702
- ContinuousOnstatement and proof · cited by 1,411
- Set.Ioostatement and proof · cited by 1,214
- derivstatement and proof · cited by 676
- IsOpen.mem_nhdsproof · cited by 470
- DifferentiableOnstatement and proof · cited by 419
- DifferentiableAt.hasDerivAtproof · cited by 114
- isOpen_Iooproof · cited by 54
- DifferentiableWithinAt.differentiableAtproof · cited by 21
- exists_hasDerivAt_eq_slopeproof · cited by 2
Cited by6
Results whose statement or proof uses this declaration.
- MonotoneOn.convexOn_of_derivproof · cited by 4
- Convex.mul_sub_le_image_sub_of_le_derivproof · cited by 3
- Convex.mul_sub_lt_image_sub_of_lt_derivproof · cited by 3
- StrictMonoOn.exists_slope_lt_deriv_auxproof · cited by 1
- exists_deriv_eq_slope'proof · cited by 1
- StrictMonoOn.exists_deriv_lt_slope_auxproof · cited by 1