Quiz interactif : Maîtrisez l’étude des fonctions numériques !

Question 1 sur 10 10:00

[{"id":68922,"question":"Soit f(x) = (2x + 1)\/(x - 3). Quel est le domaine de définition de f ?","option_a":"R","option_b":"R \\ {3}","option_c":"R \\ {-3}","option_d":"R+","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le domaine de définition exclut les valeurs qui annulent le dénominateur. Ici, x - 3 = 0 ⇒ x = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68923,"question":"La fonction f(x) = x² + 2x - 3 est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une fonction paire vérifie f(-x) = f(x). Ici, f(-x) = x² - 2x - 3 ≠ f(x). Donc elle n’est pas paire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68924,"question":"Quelle est la dérivée de f(x) = 3x⁴ - 2x² + 5 ?","option_a":"12x³ - 4x","option_b":"12x³ - 4x + 5","option_c":"3x³ - 2x","option_d":"6x³ - 4x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de 3x⁴ est 12x³, celle de -2x² est -4x, et la dérivée de 5 est 0. Donc f'(x) = 12x³ - 4x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68925,"question":"Si f'(x) \u003E 0 pour tout x ∈ I, alors f est strictement croissante sur I.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Si la dérivée est strictement positive sur un intervalle, la fonction est strictement croissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68926,"question":"Quelle est la limite de f(x) = (x² - 1)\/(x - 1) quand x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"C","explication":"En factorisant, f(x) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. Donc lim(x→1) f(x) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68927,"question":"La fonction f(x) = x³ - 3x + 2 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":"f'(x) = 3x² - 3. f'(1) = 0 et f''(1) = 6 \u003E 0, donc f admet un minimum local en x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68928,"question":"Quelle est l’asymptote horizontale de f(x) = (2x² + 1)\/(x² - 4) quand x tend vers ±∞ ?","option_a":"y = 0","option_b":"y = 2","option_c":"y = -2","option_d":"Aucune","option_e":"","option_f":"","bonne_reponse":"B","explication":"En divisant numérateur et dénominateur par x², on obtient f(x) ≈ 2\/(1 - 4\/x²) → 2 quand x → ±∞. Donc l’asymptote est y = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68929,"question":"Si f est dérivable en a et f'(a) = 0, alors f admet un extremum en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"f'(a) = 0 est une condition nécessaire mais pas suffisante. Il faut aussi étudier le signe de f' autour de a ou utiliser la dérivée seconde.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68930,"question":"Quelle est la dérivée de f(x) = √(x² + 4) ?","option_a":"x\/√(x² + 4)","option_b":"(x² + 4)\/√(x² + 4)","option_c":"2x\/√(x² + 4)","option_d":"1\/√(x² + 4)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la formule de dérivation de √u, f'(x) = (2x)\/(2√(x² + 4)) = x\/√(x² + 4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":68931,"question":"La fonction f(x) = e^x 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 e^x est e^x, qui est toujours positive sur R. Donc f est strictement croissante sur R.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]