Maîtrisez l'étude des fonctions en Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":8333,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"x + 3","option_c":"2x² + 3","option_d":"3x - 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de dérivation terme à terme : f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8334,"question":"La fonction f(x) = x³ - 3x admet-elle des points critiques ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les points critiques sont obtenus en résolvant f'(x) = 0. Ici, f'(x) = 3x² - 3, donc x = ±1 sont des points critiques.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8335,"question":"Quelle est la limite de f(x) = (2x² + 1)\/(x² - 4) quand x tend vers l'infini ?","option_a":"2","option_b":"0","option_c":"1\/2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"C","explication":"En divisant numérateur et dénominateur par x², on obtient lim (2 + 1\/x²)\/(1 - 4\/x²) = 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8336,"question":"La fonction f(x) = 1\/x est-elle continue en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction 1\/x n'est pas définie en x = 0, donc elle n'est pas continue en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8337,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² en x = 1 ?","option_a":"y = 2x - 1","option_b":"y = x + 1","option_c":"y = 2x + 1","option_d":"y = x²","option_e":"","option_f":"","bonne_reponse":"A","explication":"La tangente a pour équation y = f'(a)(x - a) + f(a). Ici, f'(1) = 2 et f(1) = 1, donc y = 2(x - 1) + 1 = 2x - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8338,"question":"La fonction f(x) = x^4 - 4x³ admet-elle un minimum local en x = 3 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) = 4x³ - 12x² et f''(x) = 12x² - 24x. En x = 3, f'(3) = 0 et f''(3) = 36 \u003E 0, donc c'est un minimum local.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8339,"question":"Quelle est l'asymptote verticale de la fonction f(x) = (x² + 1)\/(x - 2) ?","option_a":"x = 2","option_b":"x = 0","option_c":"y = 2","option_d":"y = x + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction n'est pas définie en x = 2, donc x = 2 est une asymptote verticale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8340,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction valeur absolue n'est pas dérivable en x = 0 car elle présente un point anguleux (dérivée à gauche ≠ dérivée à droite).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8341,"question":"Quelle est la dérivée seconde de f(x) = x³ - 2x² + x ?","option_a":"6x - 4","option_b":"3x² - 4x + 1","option_c":"6x² - 4","option_d":"x² - 2x + 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) = 3x² - 4x + 1, donc f''(x) = 6x - 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":8342,"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 f'(x) = e^x est toujours positive, donc la fonction est strictement croissante sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]