Quiz — Continuité sur un intervalle.pdf

Question 1 sur 10 20:00

[{"id":64926,"question":"Soit f une fonction définie sur ℝ telle que f(2) = 3. Si lim(x→2) f(x) = 3, alors :","option_a":"f est continue en 2","option_b":"f est discontinue en 2","option_c":"On ne peut pas conclure","option_d":"f est dérivable en 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction est continue en 2 car la limite en ce point existe et est égale à f(2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f est continue en 2\", \"b\": \"f est discontinue en 2\", \"c\": \"On ne ","_debug_options_count":4},{"id":64927,"question":"Le théorème des valeurs intermédiaires garantit l'existence d'une solution à l'équation f(x) = k pour toute valeur k comprise entre f(a) et f(b).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le TVI s'applique uniquement si f est continue sur l'intervalle [a, b].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":64928,"question":"La fonction f(x) = 1\/x est continue sur :","option_a":"ℝ","option_b":"ℝ*","option_c":"]-∞,0[","option_d":"[0,+∞[","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction est discontinue en 0, donc elle n'est pas continue sur ℝ ni sur [0,+∞[. Elle est continue sur ℝ* (tous les réels sauf 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"ℝ\", \"b\": \"ℝ*\", \"c\": \"]-∞,0[\", \"d\": \"[0,+∞[\"}}","_debug_options_count":4},{"id":64929,"question":"Si f est continue sur [a, b] et f(a) \u003C 0 \u003C f(b), alors l'équation f(x) = 0 admet au moins une solution dans [a, b].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une application directe du théorème des valeurs intermédiaires.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":64930,"question":"La fonction f(x) = x² - 4x + 3 est continue sur ℝ car :","option_a":"C'est une fonction polynôme","option_b":"Elle est dérivable","option_c":"Elle est bornée","option_d":"Elle est monotone","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toutes les fonctions polynômes sont continues sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"C'est une fonction polynôme\", \"b\": \"Elle est dérivable\", \"c\": \"","_debug_options_count":4},{"id":64931,"question":"Soit f une fonction continue sur [0, 5] avec f(0) = -2 et f(5) = 3. Alors :","option_a":"f(x) = 0 n'a pas de solution dans [0, 5]","option_b":"f(x) = 0 a au moins une solution dans [0, 5]","option_c":"On ne peut pas conclure","option_d":"f est constante sur [0, 5]","option_e":"","option_f":"","bonne_reponse":"b","explication":"D'après le TVI, comme f(0) \u003C 0 \u003C f(5), il existe au moins un c dans [0, 5] tel que f(c) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f(x) = 0 n'a pas de solution dans [0, 5]\", \"b\": \"f(x) = 0 a au mo","_debug_options_count":4},{"id":64932,"question":"La fonction f(x) = |x| est continue en 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction valeur absolue est continue en 0 car lim(x→0) |x| = |0| = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":64933,"question":"Si f est continue sur [a, b], alors elle est bornée sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une conséquence du théorème des bornes atteintes pour les fonctions continues sur un segment.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":64934,"question":"La fonction f(x) = sin(x) est continue sur :","option_a":"ℝ","option_b":"[0, π]","option_c":"]-∞,0]","option_d":"ℝ+","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction sinus est continue sur tous les réels ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"ℝ\", \"b\": \"[0, π]\", \"c\": \"]-∞,0]\", \"d\": \"ℝ+\"}}","_debug_options_count":4},{"id":64935,"question":"Soit f une fonction continue sur [1, 4] avec f(1) = 5 et f(4) = -1. Alors l'équation f(x) = 2 admet :","option_a":"Aucune solution","option_b":"Exactement une solution","option_c":"Au moins une solution","option_d":"Deux solutions","option_e":"","option_f":"","bonne_reponse":"c","explication":"D'après le TVI, comme 2 est compris entre f(1) et f(4), il existe au moins une solution dans [1, 4].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Aucune solution\", \"b\": \"Exactement une solution\", \"c\": \"Au moins ","_debug_options_count":4}]