Quiz — Serie 1 Revision DS2.pdf

Question 1 sur 10 20:00

[{"id":41935,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 5x - 2 ?","option_a":"f'(x) = 6x + 5","option_b":"f'(x) = 3x + 5","option_c":"f'(x) = 6x² + 5","option_d":"f'(x) = 6x + 5x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (3x²)' = 6x et (5x)' = 5, donc f'(x) = 6x + 5.","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'(x) = 6x + 5\", \"b\": \"f'(x) = 3x + 5\", \"c\": \"f'(x) = 6x² + 5\", ","_debug_options_count":4},{"id":41936,"question":"La fonction f(x) = x³ - 3x admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En x = 1, f'(1) = 3(1)² - 3 = 0. De plus, f''(1) = 6(1) = 6 \u003E 0, donc la fonction admet un minimum local en x = 1.","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":41937,"question":"Quelle est la dérivée de la fonction f(x) = (2x + 1)(x² - 3) ?","option_a":"f'(x) = 2(x² - 3) + (2x + 1)(2x)","option_b":"f'(x) = (2x + 1)'(x² - 3) + (2x + 1)(x² - 3)'","option_c":"f'(x) = 4x + 2x² - 6","option_d":"f'(x) = 2x² + x - 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la règle de dérivation d'un produit : (uv)' = u'v + uv'. Ici, u = 2x + 1 et v = x² - 3, donc f'(x) = 2(x² - 3) + (2x + 1)(2x).","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'(x) = 2(x² - 3) + (2x + 1)(2x)\", \"b\": \"f'(x) = (2x + 1)'(x² -","_debug_options_count":4},{"id":41938,"question":"La fonction f(x) = x⁴ - 4x³ est convexe sur l'intervalle [0 ; 3] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La convexité est déterminée par le signe de f''(x). Ici, f''(x) = 12x² - 24x = 12x(x - 2). Sur [0 ; 3], f''(x) change de signe (négatif sur [0 ; 2], positif sur [2 ; 3]), donc la fonction n'est pas convexe sur tout l'intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41939,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"f'(x) = 2cos(2x)","option_b":"f'(x) = cos(2x)","option_c":"f'(x) = 2sin(x)","option_d":"f'(x) = -2cos(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la règle de dérivation des fonctions composées : (sin(u))' = u'cos(u). Ici, u = 2x, donc u' = 2 et f'(x) = 2cos(2x).","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'(x) = 2cos(2x)\", \"b\": \"f'(x) = cos(2x)\", \"c\": \"f'(x) = 2sin(x)\"","_debug_options_count":4},{"id":41940,"question":"La fonction f(x) = e^(3x) est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = e^(3x) est f'(x) = 3e^(3x). Comme e^(3x) \u003E 0 pour tout x ∈ ℝ, f'(x) \u003E 0, donc la fonction est strictement croissante 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41941,"question":"Quelle est la valeur de la dérivée de f(x) = ln(x² + 1) en x = 1 ?","option_a":"f'(1) = 1","option_b":"f'(1) = 0.5","option_c":"f'(1) = 2","option_d":"f'(1) = -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de f(x) = ln(u) est f'(x) = u'\/u avec u = x² + 1, donc u' = 2x. Ainsi, f'(x) = 2x\/(x² + 1). En x = 1, f'(1) = 2\/(1 + 1) = 1.","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'(1) = 1\", \"b\": \"f'(1) = 0.5\", \"c\": \"f'(1) = 2\", \"d\": \"f'(1) = -","_debug_options_count":4},{"id":41942,"question":"La fonction f(x) = x³ - 6x² + 9x admet-elle un point d'inflexion en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un point d'inflexion se produit lorsque f''(x) = 0 et change de signe. Ici, f''(x) = 6x - 12. En x = 2, f''(2) = 0 et f'' change de signe (négatif avant 2, positif après), donc x = 2 est un point d'inflexion.","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":41943,"question":"Quelle est la dérivée de la fonction f(x) = (x² + 1)\/(x - 1) ?","option_a":"f'(x) = (2x(x - 1) - (x² + 1))\/(x - 1)²","option_b":"f'(x) = (2x)\/(x - 1)","option_c":"f'(x) = (x² + 1 - 2x)\/(x - 1)²","option_d":"f'(x) = (2x(x - 1) + (x² + 1))\/(x - 1)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la règle de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = x² + 1 et v = x - 1, donc f'(x) = (2x(x - 1) - (x² + 1)(1))\/(x - 1)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'(x) = (2x(x - 1) - (x² + 1))\/(x - 1)²\", \"b\": \"f'(x) = (2x)\/(x","_debug_options_count":4},{"id":41944,"question":"La fonction f(x) = x⁴ - 4x + 3 admet-elle un minimum global en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 4x³ - 4 s'annule en x = 1 (f'(1) = 0). De plus, f''(x) = 12x² \u003E 0 pour tout x, donc la fonction est convexe et x = 1 est un minimum global.","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}]