Testez vos connaissances sur l'égalité des accroissements finis

Question 1 sur 10 10:00

[{"id":28418,"question":"Soit f une fonction dérivable sur [a,b]. Que garantit le théorème des accroissements finis ?","option_a":"Il existe c ∈ ]a,b[ tel que f'(c) = (f(b)-f(a))\/(b-a)","option_b":"La fonction f est constante sur [a,b]","option_c":"La dérivée f' est bornée sur [a,b]","option_d":"La fonction f est convexe sur [a,b]","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le théorème des accroissements finis affirme qu'il existe au moins un point c dans l'intervalle ouvert où la dérivée de f est égale à la pente de la sécante entre a et b.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28419,"question":"Le théorème des accroissements finis s'applique-t-il à la fonction f(x) = |x| sur l'intervalle [-1,1] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Non, car la fonction |x| n'est pas dérivable en x=0, qui appartient à l'intervalle [-1,1].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28420,"question":"Si f'(x) = 2x pour tout x ∈ ℝ, que vaut f(3)-f(1) ?","option_a":"4","option_b":"8","option_c":"12","option_d":"16","option_e":"","option_f":"","bonne_reponse":"C","explication":"Par le théorème des accroissements finis, f(3)-f(1) = f'(c)(3-1) = 2c×2. Comme f'(x)=2x, c=2 (unique solution), donc f(3)-f(1)=8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28421,"question":"Soit f(x) = x². Quel est le point c garanti par le théorème des accroissements finis sur [1,3] ?","option_a":"c=1","option_b":"c=2","option_c":"c=3","option_d":"c=4","option_e":"","option_f":"","bonne_reponse":"B","explication":"f'(x)=2x. Le théorème donne f'(c) = (9-1)\/(3-1)=4 ⇒ 2c=4 ⇒ c=2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28422,"question":"Une fonction dont la dérivée est toujours positive est-elle strictement croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Faux. Exemple : f(x)=x³ a f'(x)=3x² ≥ 0 mais n'est pas strictement croissante (plate en x=0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28423,"question":"Soit f(x) = sin(x) sur [0, π\/2]. Que vaut f'(c) pour le c du théorème des accroissements finis ?","option_a":"0","option_b":"1","option_c":"π\/2","option_d":"2\/π","option_e":"","option_f":"","bonne_reponse":"B","explication":"f(π\/2)-f(0)=1-0=1. La pente de la sécante est 1\/(π\/2)=2\/π. f'(c)=cos(c)=2\/π ⇒ c=arccos(2\/π) ≈ 0.88 rad. Mais cos(c)=2\/π ≈ 0.636, donc f'(c)=0.636 (pas 1). Correction : la bonne réponse est 2\/π (option D).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28424,"question":"Le théorème des accroissements finis permet-il de majorer |f(b)-f(a)| ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Vrai, si on connaît un majorant de |f'(x)| sur [a,b], alors |f(b)-f(a)| ≤ M|b-a| où M est ce majorant.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28425,"question":"Soit f(x) = x³ - 3x sur [-2,2]. Quel est le maximum de |f'(x)| sur cet intervalle ?","option_a":"3","option_b":"6","option_c":"9","option_d":"12","option_e":"","option_f":"","bonne_reponse":"C","explication":"f'(x)=3x²-3. |f'(x)| est maximale en x=±2 : |f'(2)|=|12-3|=9.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28426,"question":"Si f'(x) ≤ 5 pour tout x ∈ [0,1], que peut-on dire de f(1)-f(0) ?","option_a":"f(1)-f(0) ≤ 5","option_b":"f(1)-f(0) ≥ 5","option_c":"0 ≤ f(1)-f(0) ≤ 5","option_d":"|f(1)-f(0)| ≤ 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par le théorème des accroissements finis, f(1)-f(0)=f'(c)(1-0)=f'(c) ≤ 5 car f'(c) ≤ 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":28427,"question":"Soit f(x) = eˣ. Le point c du théorème des accroissements finis sur [0,1] vérifie :","option_a":"c = 0","option_b":"c = 1","option_c":"c = ln(1)","option_d":"c = ln(e-1)","option_e":"","option_f":"","bonne_reponse":"D","explication":"f(1)-f(0)=e-1. f'(c)=eᶜ=e-1 ⇒ c=ln(e-1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]