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[{"id":65302,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"x² + 3","option_c":"2x - 5","option_d":"x + 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle : (x^n)' = n*x^(n-1). Ici, f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65303,"question":"Si une fonction est dérivable en un point, alors elle est nécessairement continue en ce point.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est un théorème fondamental : la dérivabilité implique la continuité, mais l'inverse n'est pas toujours vrai.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65304,"question":"Quelle est l'équation de l'asymptote verticale de la fonction f(x) = (2x + 1)\/(x - 3) ?","option_a":"x = 3","option_b":"y = 2","option_c":"x = -3","option_d":"y = x - 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'asymptote verticale s'obtient en annulant le dénominateur : x - 3 = 0 ⇒ x = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65305,"question":"La fonction f(x) = x³ - 3x² a-t-elle un extremum local en x = 2 ?","option_a":"Oui, un maximum","option_b":"Oui, un minimum","option_c":"Non","option_d":"Oui, un point d'inflexion","option_e":"","option_f":"","bonne_reponse":"B","explication":"f'(x) = 3x² - 6x. f'(2) = 0 et f''(2) = 6 \u003E 0 ⇒ minimum local en x = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65306,"question":"Si lim(x→+∞) f(x) = -∞ et lim(x→-∞) f(x) = +∞, la fonction f admet-elle nécessairement une asymptote horizontale ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une asymptote horizontale existe uniquement si les limites en ±∞ sont finies et égales. Ici, les limites sont infinies, donc pas d'asymptote horizontale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65307,"question":"Quelle est la limite de la fonction f(x) = (sin x)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est une limite classique : lim(x→0) (sin x)\/x = 1 (théorème des gendarmes ou développement limité).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65308,"question":"Une fonction peut-elle avoir deux asymptotes horizontales différentes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Oui, si lim(x→+∞) f(x) = L1 et lim(x→-∞) f(x) = L2 avec L1 ≠ L2, alors la fonction admet deux asymptotes horizontales distinctes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65309,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = x² en x = 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"4","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le coefficient directeur de la tangente est donné par f'(1). f'(x) = 2x ⇒ f'(1) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65310,"question":"La fonction f(x) = e^x est-elle toujours 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 sur ℝ, donc la fonction est strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65311,"question":"Quelle est la nature de l'extremum de la fonction f(x) = -x⁴ + 4x² en x = √2 ?","option_a":"Maximum local","option_b":"Minimum local","option_c":"Point d'inflexion","option_d":"Aucun extremum","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) = -4x³ + 8x. f'(√2) = 0 et f''(√2) = -8√2 \u003C 0 ⇒ maximum local en x = √2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]