(** ** Questions 2 *)
(* 8:38 *)
(** 2.a. *)

(* Specifications: *)
Lemma Signe_df : forall x, 0 < x -> sign (Derive f x) = sign (- ln x).
(** 2.b. *)
Lemma filterlim_f_0 :
filterlim f (at_right 0) (Rbar_locally m_infty).
Lemma Lim_f_p_infty : is_lim f p_infty 0.
(** 2.c. *)
Lemma Variation_1 : forall x y, 0 < x -> x < y -> y < 1 -> f x < f y.
Lemma Variation_2 : forall x y, 1 < x -> x < y -> f x > f y.
(** ** Questions 3 *)
(* 9:40 *)
(** 3.a *)
Lemma f_eq_1_0_1 : exists x, 0 < x <= 1 /\ f x = 1.
(** 3.b. *)
Lemma f_eq_1_1_p_infty : exists x, 1 <= x /\ f x = 1.
(** ** Questions 5 *)
(* 10:08 *)
(** 5.a. *)
(** 5.b. *)
Lemma If : forall x, 0 < x -> is_derive (fun y : R => 2 * ln y + (ln y) ^ 2) x (f x).
Lemma RInt_f : is_RInt f ( / exp 1) 1 1.
Fixpoint u (n : nat) : R :=
match n with
| O => 2
| S n => 2/3 * u n + 1/3 * (INR n) + 1
end.