Grade 12 / Additional information on derivation and convexity 51 exercises (including 50 corrected)

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1. Derivatives of compound functions E.3506 Determine the expression of the deriva-tive of each of the following functions : E.5103 Determine the expression of the deriva-tives of the following functions : a f ( x ) = 6 x 2 − 3 x + 1 b g ( x ) = 4 · x − 1 4 E.95 Consider the function f defined by: f : x ↦−→ − 2 x 2 − x + 6 1 Determine the domain of the function f . 2 Give the expression of the derivative of the function f . E.5217 Consider the function f defined on the interval −∞ ; − 1 2 by the relationship through the following relationship : f ( x ) = 2 x 2 − 3 x − 2 Determine the slope-intercept formof the tangent to the curve C f in − 1 E.5751 Determine the expression of the deriva-tives of the following functions : a f ( x ) = 3 x + 2 5 b g ( x ) = 1 − 2 x c h ( x ) = 1 x 2 + x + 1 d j ( x ) = x x + 1 E.2394 1 Determine the derivative function of the function f de-fined by: f : x ↦−→ 5 · x 2 + x + 1 4 2 Determine the derivative function of the function g de-fined by: g : x ↦−→ 2 5 x 2 · 3 − 2 x Write this function derived from the function g as a quo-tient whose denominator is 3 − 2 x . E.6137 Consider the function f defined on R by the relation: f ( x ) = a · x 2 + b · x 4 · c · x + 1 où a , b and c are three relative integers. We know that the function f admits as derivative the func-tion f whose expression is : f ( x ) = − 18 x 2 + 21 x − 4 · a · x 2 + b · x 3 Determine the expression of the function f . E.8399 Determine the expression, in simplified form, of the function f derived from the function f defined by: f ( x ) = x + 1 3 E.8400 For each of the functions below, give the simplified expression of their derivative function : a f ( x ) = 3 x 2 − 2 x +1 x b g ( x ) = (2 x +1) · 3 − x 2. Derivatives of compound functions of the exponential function E.8410 For each function, determine the expres-sion of the function f derived from the function f : a f ( x ) = e 2 − x 2 b f ( x ) = e x 2 +1 c f ( x ) = e x 2 + x +1 E.8420 For each function, determine the expres-sion of the function f derived from the function f : a f ( x ) = 2 · x + 1 e x +1 b f ( x ) = x · e 3 · x 2 E.3613 Determine the expression of the deriva-tive functions of each of the following functions : 1 f ( x ) = e √ x 2 g ( x ) = x · e x 2 +1 3 h ( x ) = x · e − 1 x 4 j ( x ) = e − 2 x +1 2 x + 1 5 k ( x ) = e x 2 + x 6 ‘ ( x ) = e x 2 +1 E.5140 The table below shows functions and their derivatives. Check the accuracy of the table : f ( x ) f ( x ) a 3 x · e 5 x 2 +3 30 x 2 + 3 · e 5 x 2 +3 b e 3 − x 2 · e x 2 +1 0 c e 2 − x 2 2 x − 2 x 2 + 1 e 2 − x 2 2 x 2 d e x √ x 3 2 · √ x · e x √ x e e 2 x 2 − x 4 ( x − 2) 2 · e 2 x 2 − x 3. Derivatives of function families E.6134 Consider, for any non-zero natural num-ber n , the function f n defined on R by the relation: f n ( x ) = 1 ( x 2 + 1) n In a reference frame O ; I ; J orthonormal, note C n the rep-resentative curve of the function f n . https://chingmath.fr chapExoCorrec/3506 sacados/3506 chapExoCorrec/5103 sacados/5103 chapExoCorrec/95 sacados/95 chapExoCorrec/5217 sacados/5217 chapExoCorrec/5751 sacados/5751 chapExoCorrec/2394 sacados/2394 chapExoCorrec/6137 sacados/6137 chapExoCorrec/8399 sacados/8399 chapExoCorrec/8400 sacados/8400 chapExoCorrec/8410 sacados/8410 chapExoCorrec/8420 sacados/8420 chapExoCorrec/3613 sacados/3613 chapExoCorrec/5140 sacados/5140 chapExoCorrec/6134 sacados/6134
-2-12IJOC1C3C8 IJOC1C3C2 -3-2-123I-12JOCfCg 1 Determine the expression of the derivative f n of the func-tion f n . 2 a Determine the slope-intercept formof the tangent ( d ) to the curve C 3 at the point of abscissa 1 . b Consider the straight line (Δ) of slope-intercept form y = 5 32 · x + 3 16 Of the family of curves C n , which admits the straight line (Δ) as tangent at the point of abscissa − 1 ? Justify your answer. E.6810 Consider for any non-zero natural num-ber n , the function f n defined on R by the relation: f n ( x ) = 2 · x + n 1 + x 2 In a reference frame O ; I ; J orthonormal, note C n the rep-resentative curve of the function f n . 1 Determine the expression of the derivative function f n . 2 Determine the slope-intercept formof the tangent ( d n ) to the curve C n at the point of abscissa 1 . 3 a Establish the following equality: n · x 3 − 2 · n +1 · x 2 + n +2 · x − 1 = x − 1 2 · n · x − 1 b Study the relative position of the line d n and the curve C n for any natural number n greater than or equal to 2 . 4. Tangent of compound functions E.7740 Consider the function g defined by the relation: g ( x ) = x 2 − 2 x + 3 Note C g the representative curve of the function g . 1 Determine the definition set of the function g . 2 Determine the equation of the tangent to the curve C g at the point of abscissa 3 2 . E.6919 Consider the function f de-fined for any real x by: f ( x ) = x · e 1 − x 2 and the function g defined for any real x by: g ( x ) = e 1 − x On the graph below, the representative curves C f and C g of the functions f and g respectively have been plotted on a reference frame. Justify that at the point of abscissa 1 , the curves C f and C g admit the same tangent. 5. Table of variations of a compound function E.5752 Consider the function f defined by the relation: f ( x ) = 6 x 2 + 13 x − 5 1 Justify that the function f admits as definition set the part I of R defined by: I = −∞ ; − 5 2 ∪ 1 3 ; + ∞ . 2 Determine the direction of variation of the function f on its defining set I . https://chingmath.fr -2-12IJOC1C3C8 chapExoCorrec/6810 sacados/6810 IJOC1C3C2 chapExoCorrec/7740 sacados/7740 chapExoCorrec/6919 sacados/6919 Extrait d'Antilles-Guyane Juin 2016 -3-2-123I-12JOCfCg chapExoCorrec/5752 sacados/5752
-3-2-123I-2-1JOCf -4-3-2-1234I23JOCf E.6135 Consider the function f defined on R by the relation: f ( x ) = x 2 + 2 x + 5 x 2 + 1 1 Establish that the derivative f of the function f admits as expression : f ( x ) = ( x − 1)( x 2 + x − 2 x 2 + 1 · x 2 + 1 2 Draw up the table of variations of the function f . We admit the two limits: lim x ↦→−∞ f ( x ) = + ∞ ; lim x ↦→ + ∞ f ( x ) = + ∞ 6. Study the derivative function of compound functions E.5753 Consider the function f whose image of a number x ∈ R is defined by the relation: f ( x ) = 1 8 · x 2 − x − 2 3 Below is the representative curve C f of the function f in an orthonormal coordinate system O ; I ; J . 1 Draw up the table of variations of the function f on R . 2 Determine the equation of the tangent to the curve C f at the point with abscissa 1 . (We will use the approximate value f ( 1 2 ≈− 1 ; 4 ) E.5062 Consider the function f defined on R whose image of a number x is given by the relation: f ( x ) = (5 x 2 + 3 x + 2) 5 1 Determine the expression of the derivative f of the func-tion f . 2 Study the variations of the function f on R . E.5063 Consider the function f whose image of a number x is given by the relation: f ( x ) = 2 x 2 + x + 1 1 Determine the definition set of the function f . 2 Determine the table of variations of the function f . 3 In a reference frame O ; I ; J , we give the curve C f rep-resentative of the function f : a Determine the equation of the tangent ( d ) to the curve C f at the point of abscissa 1 . b Draw the straight line ( d ) in the above reference frame. https://chingmath.fr chapExoCorrec/6135 sacados/6135 chapExoCorrec/5753 sacados/5753 -3-2-123I-2-1JOCf chapExoCorrec/5062 sacados/5062 chapExoCorrec/5063 sacados/5063 -4-3-2-1234I23JOCf
DBM3km15kmB IJO4BCf 234567I234567JO E.2962 A boat, represented by the point B , is three kilometers from shore ; the point B represents the point on the shore closest to the boat (its orthogonal pro-jected) ; the point D represents the destination to be reached by the sailor. To do this, the sailor decides to reach a point M from the shore, then drive to the point D . The boat sails at a speed of 15 km = h and the car moves at a speed of 40 km = h . We note x the distance B M : 1 Express the distance BM and MD as a function of x . 2 Note h ( x ) the travel time taken when B M = x . a Justify that : h ( x )= 1 120 · 8 x 2 +9+45 − 3 x b Determine the expression of the derivative function h . c Draw up the table of variations of the function h . 3 Deduce the value of x for which the sailor’s travel time is minimal. E.6163 Consider the function f defined by the relation: f ( x ) = ( a · x + 1) · 2 x 2 + x + 1 2 In an orthogonal O ; I ; J given below, we represent the curve C f representative of the function f : The line ( d ) passes through the points J and B (1 ; 4) . 1 a Justify that the curve C f passes through the point J . b Determine the coefficient of the line ( JB ) . c Demonstrate that for any real x , we have : f ( x ) = 10 ax 2 + (3 a + 8) · x + ( a + 2) · 2 x 2 + x + 1 d Assume that the straight line ( JB ) is tangent to the curve C f at the point J . Determine the value of a . Justify your answer. 2 It is assumed that f has the expression : f ( x ) = 10 x 2 + 11 x + 3 2 x 2 + x + 1 Determine the directions of variation of the function f on R . E.3643 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = 1 x 2 · e 1 x Note C the representative curve of the function f in an or-thonormal reference frame O ; −→ i ; −→ j . The graphic unit is 1 cm . 1 Limit study a Determine the limit of the function f when x tends to 0 . b Determine the limit of the function f when x tends to + ∞ . c What consequences can be deduced from these two re-sults, for the curve C ? 2 Study the variations of the function f . a Demonstrate that, the derivative function of the func-tion f is expressed, for any strictly positive real x , by: f ( x ) = − 1 x 4 · e 1 x · (2 x + 1) b Determine the sign of f and deduce the table of vari-ations of f on the interval 0 ; + ∞ . c Demonstrate that the equation f ( x )=2 has a unique solution noted ¸ belonging to the interval 0 ; + ∞ and give the approximate value of ¸ rounded to the hun-dredth. 3 Draw the curve C in the orthonormal reference frame O ; −→ i ; −→ j . 7. Link between derivative and derivative number https://chingmath.fr chapExoCorrec/2962 sacados/2962 Plutot pour les terminales DBM3km15kmB chapExoCorrec/6163 sacados/6163 IJO4BCf chapExoCorrec/3643 sacados/3643 Extrait Juin 2010 234567I234567JO
23I-123JOCk 05101520253035404550556024681012141618C E.5064 Determine the value of the following limits: a lim h ↦→ 0 3 + h 4 − 81 h b lim h ↦→ 0 6 · h + 1 − 1 h c lim x ↦→ 0 x 2 + 1 8 − 1 x d lim x ↦→ 0 x 2 + x + 4 − 2 x E.5101 Determine the values of the following limits: a lim h ↦→ 0 2 h + 5 − √ 5 h b lim h ↦→ 0 (2 − 2 h ) 4 − 16 h E.5829 Let f be a derivable numerical function. 1 Let a ∈D f . Establish the following limit: lim x ↦→ a x = a x · f ( a ) − a · f ( x ) x − a = f ( a ) − a · f ( a ) Let’s put : x = a + h . 2 Deduce the limit: lim x ↦→ a x = a x 3 · a − a 3 · x x − a 8. Convexity: graphically E.7011 Plotted below is the graphical rep-resentation of the second derivative k  of a function k defined on 0 ; + ∞ . Which of the following answers is correct? 1 k is concave on the interval 1 ; 2 . 2 k is convex on the interval 0 ; 2 . 3 k is convex on 0 ; + ∞ . 4 k is concave at 0 ; + ∞ . E.7059 Consider a function P defined and derivable on the interval 0 ; 60 . The representative curve C of the function P is given below. From a graphical reading answer the following questions : 1 Arguing the answer, give the sign of P (54) , où P is the derivative function of the function P . 2 Give an interval on which the function P is convex. 3 Give, to the nearest unit, the solutions of the equation : P ( x )=10 . https://chingmath.fr chapExoCorrec/5064 sacados/5064 chapExoCorrec/5101 sacados/5101 chapExoCorrec/5829 sacados/5829 Bac Maroc Septembre 1969 chapExoCorrec/7011 sacados/7011 Extrait de Liban Mai 2016 23I-123JOCk chapExoCorrec/7059 sacados/7059 05101520253035404550556024681012141618C
-3-2-1234567I-6-5-4-3-2-1234JOC3C1C2 E.7054 In the orthogonal reference frame below three curves C 1 , C 2 and C 3 defined on − 3 ; 7 were represented. One of these represents a f function, another represents its derivative and a third represents its second derivative. Explain how these graphical representations can be used to determine the convexity of the function f . Indicate an interval on which the function f is convex. 9. Convexity: study of functions E.7730 Which of the following statements is correct. The function g defined on R by g ( x )= x 3 − 9 · x is convex on the interval: a −∞ ; + ∞ b 0 ; + ∞ c −∞ ; 0 d − 3 ; 3 E.7702 For the question asked, only one of the four answers is correct. No justification is required. The function f is defined for any real number x belonging to 0 ; + ∞ by: f ( x ) = e 2 · x +ln( x ) 1 function f is concave. 2 function f has a second derivative function that cancels. 3 function f has a strictly positive second derivative func-tion. 4 function f has a derivative function that cancels. E.7043 Let f be the function defined on R by: f ( x ) = x · e x 2 − 1 C f is the representative curve of the function f in an orthonor-mal plane. Let f be the derivative function of f and f  the second derivative function of f . 1 a Show that for any real x : f ( x ) = 2 · x 2 + 1 · e x 2 − 1 b Deduce the direction of variation of f at R . 2 We admit that for any real x : f  ( x ) = 2 x · 2 · x 2 + 3 · e x 2 − 1 Determine, with justification, the interval on which the function f is convex. E.7793 Consider the function f defined, for any real x in the interval − 2 ; 4 by: f ( x )= x +2 · e − x +1 Let f be the derivative function of f . 1 Show that, for any x in the interval − 2 ; 4 , we have : f ( x ) = − x + 1 · e − x +1 2 Formal calculation software gives the following results : 1 factoriser dériver − ( x +1) ∗ exp( − x +1) x ∗ exp − x + 1 2 intégrer x +2 ∗ exp( − x +1) − ( x + 3) ∗ exp − x + 1 Using these results, answer the following question : Determine an interval on which the function f is convex. Justify. E.7026 Let f be the function defined on 0 ; 8 by: f ( x )= 0.4 20 · e − x +1 +0.4 1 Montrer que f ( x )= 8 · e − x 20 · e − x +1 2 où f denotes the func-tion derived from the function f . 2 Formal calculation software gives the results below : 1 f’(x):=8*e^(-x)/(20*e^(-x)+1)^2 f ( x ) := 8 · e − x 400 e − x 2 + 40 · e − x + 1 2 g(x):=Dérivée f’(x) g ( x ) := 160 · e − x 2 − 8 · e − x 8000 · e − x 3 + 1200 · e − x 2 + 60 · e − x + 1 3 Factoriser g(x) 8 · e − x · 20 · e − x − 1 20 · e − x + 1 3 Based on these results, determine the interval over which the function f is convex. https://chingmath.fr chapExoCorrec/7054 sacados/7054 -3-2-1234567I-6-5-4-3-2-1234JOC3C1C2 chapExoCorrec/7730 sacados/7730 chapExoCorrec/7702 sacados/7702 chapExoCorrec/7043 sacados/7043 chapExoCorrec/7793 sacados/7793 chapExoCorrec/7026 sacados/7026
Temps (en heure)012345678910111213141516Concentration (g/‘)0,20,40,60,811,21,41,61,822,2 00,20,40,60,811,21,41,61,822,22,42,62,833,23,43,63,844,24,44,64,85-4-224(Cf(CfAB 00,511,525101520 10. Convexity and tangent positions E.7794 Consider the function f defined on the interval 0 ; 10 by: f ( x ) = − x · ln x + 2 · x + 1 Note C f the representative curve of the function f in the plane with a reference frame. Show that the curve C lies entirely below each of its tangents on the interval 0 ; 10 . E.7695 Consider the function f defined on 0 ; + ∞ by: f ( x ) = 3 · x − 3 · x · ln( x ) Note C f its representative curve in an orthonormal frame and T the tangent to C f at the point of abscissa 1 . What is the relative position of C f with respect to T ? 11. Inflection point: graphically E.7050 A patient is injected with a drug and the concentration, in grams in liters, of this drug in the blood is measured regularly for 15 hours. The curve shown below is obtained : With the help of a graph and without justification, after how many hours does the drop in concentration slow down? E.7061 Consider a function f defined on the interval 0 ; 5 . Shown belowbelow the curve C f of the derivative function f as well as the curve C f  of the second derivative function f  on the interval 0 ; 5 . The point A of coordinates (1 ; 0) belongs to C f and the point B of coordinates (2 ; 0) belongs to the curve C f  . 1 Determine the direction of variation of the function f . Justify. 2 Determine on quel (s) intervalle (s) , the function f is con-vex. Justify. 3 Does the curve of f admit points of inflection? Justify. If so, specify leur (s) abscisse (s) . 12. Inflection point: study of functions E.7020 Consider the function f defined on the interval 0 ; 15 by: f ( x )=9 · x 2 · 1 − 2 · ln x +10 . The representative curve of f is given below : https://chingmath.fr chapExoCorrec/7794 sacados/7794 chapExoCorrec/7695 sacados/7695 chapExoCorrec/7050 sacados/7050 Temps (en heure)012345678910111213141516Concentration (g/‘)0,20,40,60,811,21,41,61,822,2 chapExoCorrec/7061 sacados/7061 00,20,40,60,811,21,41,61,822,22,42,62,833,23,43,63,844,24,44,64,85-4-224(Cf(CfAB chapExoCorrec/7020 sacados/7020 00,511,525101520
Let f  ( x )= − 36 · ln x − 36 où f  denote the second derivative of the function f on the interval 0 ; 1.5 . Show that the representative curve of the function f admits an inflection point whose abscissa is e − 1 . E.7052 Monthly vegetable production will deliver at most 1 000 baskets per month. The total cost of pro-duction is modeled by the function C defined on the interval 0 ; 10 by: C ( x ) = − 1 48 · x 4 + 5 16 · x 3 + 5 · x + 10 When x is expressed in hundreds of papers, C ( x ) is equal to the total cost expressed in hundreds of euros. We admit that, for any number x of the interval 0 ; 10 , marginal cost is given by the function C m = C où C is the function derived from C . 1 Calculate C m (6) , the marginal cost for six hundred bas-kets sold. 2 We note C  the second derivative function of C and we have : C  ( x ) = − 1 4 · x 2 + 15 8 · x a Determine the largest interval of the form 0 ; a in-cluded in 0 ; 10 on which the function C is convex. b What can be said about the point of abscissa a on the curve of the function C ? Interpret this value of a in terms of cost. E.7791 Consider the function f defined on the interval − 20 ; 20 by: f ( x ) = − 2 · x + 30 · e 0.2 · x − 3 1 a Show that f ( x )= − 0.4 · x +4 · e 0.2 · x − 3 for any real x from the interval − 20 ; 20 . b Draw up the table of variations of the function f on the interval − 20 ; 20 . We will specify the exact value of the maximum of f . 2 a Show that, on the interval − 20 ; 20 , the equation f ( x )= − 2 admits a single solution ¸ . b Give a framing of ¸ of amplitude 0.2 . 3 Formal calculation software gives the results below : 1 Dériver − 10 · x +200 · e 0.2 · x +3 − 2 · x +30 · e 0.2 · x − 3 2 Dériver 2 · x +30 · e 0.2 · x − 3 − 0.4 · x +4 · e 0.2 · x − 3 3 Dériver ( − 0.4 · x +4 · e 0.2 · x − 3 − 0.08 · x +0.4 · e 0.2 · x − 3 Answer the following question using the results given by the software: Determine the largest interval on which the function f is convex and specify the abscissa of the inflection point. E.7729 We admit that the function f is defined by: f ( x ) = x 2 − 2 · x + 1 · e − 2 x +6 With the help of a calculus program, we obtain the results below, which can be used without being demonstrated : L1 f(x):=(x^2-2x+1)*e^(-2x+6) → f ( x ) = x 2 − 2 · x + 1 · e − 2 x +6 L2 f’(x):=Dérivée f(x) → f ( x ) = − 2 · x 2 + 6 · x − 4 · e − 2 x +6 L3 g(x):=Dérivée f’(x) → g ( x ) = − 16 · x · e − 2 x +6 + 4 · x 2 · e − 2 x +6 + 14 · e − 2 x +6 L4 Factoriser g(x) → 2 · e − 2 x +6 · 2 · x 2 − 8 · x + 7 L5 Résoudre g(x)=0 → x = − 2 + 4 2 ; x = 2 + 4 2 L6 F(x):=Primitive f(x) → F ( x ) = 1 4 · − 2 · x 2 + 2 · x − 1 · e − 2 · x +6 1 Determine the largest interval on which the function f is concave. 2 Does the representative curve of the function f admit points of inflection? If so, give their abscissa. E.7790 Consider the function f defined on the interval 0 ; 10 by: f ( x ) = 1 0.5 + 100 · e − x Let f be the derivative function of f on the interval 0 ; 10 . 1 Show that, for any real x in the interval 0 ; 10 , we have : f ( x ) = 100 · e − x 0.5 + 100 · e − x 2 Note f  the second derivative function of f on the interval 0 ; 10 . Formal calculation software provides the following expression for f  ( x ) : f  ( x ) = 100 · e − x · 100 · e − x − 0.5 0.5 + 100 · e − x 3 2 a Show that, in the interval 0 ; 10 , the inequation 100 · e − x − 0.5 0 is equivalent to the inequation : x − ln 0.005 . b Deduce the sign table of the function f  on the interval 0 ; 10 . 3 We call C f the representative curve of f drawn in a ref-erence frame. Show, using question 2 , that the curve C f admits a point of inflection noted I , the exact value of whose ab-scissa will be specified. 4 Using the results of question 2 , determine the interval on which the function f is concave. 13. Unclassified financial years https://chingmath.fr chapExoCorrec/7052 sacados/7052 Extrait Antilles-Guyane Septembre 2014 chapExoCorrec/7791 sacados/7791 chapExoCorrec/7729 sacados/7729 chapExoCorrec/7790 sacados/7790 Extrait Liban Juin 2017
120m77m8mdEntréeSortie E.1984 Consider the function : f : x ↦→ − 2 · x 2 + 7 · x − 3 1 Show the following equality: (2 · x − 1)(3 − x ) = − 2 · x 2 + 7 · x − 3 2 a Study the sign of − 2 · x 2 +7 · x − 3 as a function of the value of x . b Deduce the set of definition and derivability of the func-tion f . 3 Give the expression of the derivative function of the func-tion f . 4 Draw up the table of variations of the function f . E.3644 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = 6 − 5 x + 1 1 Study the direction of variation of the function f on the interval 0 ; + ∞ . 2 Solve in the interval 0 ; + ∞ , the equation f ( x )= x . Let a be the unique solution of this equation. 3 Show that if x belongs to the interval 0 ; a , then f ( x ) belongs to the interval 0 ; a . E.8215 The town ˇ Promenade ı wishes to move in a square-shaped plot crossed by a river run-ning laterally through the land. The figure below repre-sents the plot and the river is the hatched area: At what distance d should the bridge be placed so that the distance traveled by a visitor is minimal? E.111 Consider a segment [ AB ] of length 10 centimeters and a point M of this segment, different from A and B . Points N and P are such that AMNP is a square. The aim of the exercise is to determine the point M of seg-ment [ AB ] for which the distance BN is minimal. Distances are expressed in centimeters. 1 We pose : AM = x . a Make a figure. b Determine the interval of possible values for x . c Determine as a function of x the distance BM . d Determine as a function of x the distance BN . (Recall the Pythagorean theorem: in a triangle ABC right-angled A , we have BC 2 = AB 2 + AC 2 ) 2 Consider the function f defined on the interval [0 ; 10] by: f ( x ) = 2 · x 2 − 20 · x + 100 The derivative function f of f is defined on the interval [0 ; 10] by: f ( x ) = 2 x − 10 2 · x 2 − 20 · x + 100 . a Answer the following questions i Study the variations of the function f on the interval [0 ; 10] ii Show that the function f admits a minimum on the interval [0 ; 10] to be specified. b Answer the following questions : i Draw the representative curve of f in an orthonormal frame of reference with unit one centimetre. ii Graphically solve the equation f ( x )=8 . Useful con-struction lines will be shown and approximate values of the solutions read will be given. 3 Using the previous results, determine the point M of seg-ment [ AB ] for which the distance BN is minimum. https://chingmath.fr chapExoCorrec/1984 sacados/1984 chapExoCorrec/3644 sacados/3644 chapExoCorrec/8215 sacados/8215 120m77m8mdEntréeSortie sacados/111 Antilles - 2002 sept - obligatoire - 7 points
OABCIxy E.71 west Indies ffl 2004 ffl 7 points ffl Com-pulsory The figure below is a diagram of a car jack. This consists of a deformable rhombus OABC , the point O being the point of support on the ground and the point B being the point through which the car is lifted. When the crank M is turned, the nuts A and C move towards (or away) , which causes (or descends) support B to rise, along axis ( Oy ) . We give : OA = OC = AB = BC =25 cm In the orthonormal frame ( O ; x ; y ) of unit one centimetre, x A designates the abscissa of the point A and varies from 0 to 25. The ordinate of the point B is denoted y B : For x A =0 , we have : y B =50 ; For x A =25 we have : y B =0 . For the jack to work properly, you need : x A > 6 ; y B > 10 . 1 a Using the Pythagorean theorem in the AIB triangle, find a relationship between x A and y B and verify that : y B =2 · 625 − x 2 A . b Using the relationship found in question a , calculate the value of x A when y B equals 10, then the value of y B when x A equals 6. 2 Consider the function f defined for x belonging to the interval 0 ; 25 by: f ( x )=2 · 625 − x 2 . We will admit that a function of type u , where u is a function defined and positive on an interval, has the same direction of variation as u on this interval. a Determine the derivative u of the function u defined on the interval 0 ; 25 by u ( x )=625 − x 2 . Study the sign of u ( x ) when x varies between 0 and 25. Deduce the direction of variation of the function u on the interval 0 ; 25 . b Deduce the table of variations of the function f . Spec-ify the values of the function at the limits of the inter-val. c Draw the curve (Γ) representative of the function f in an orthonormal frame of reference with graphical unit 0.5 cm . (The abscissa points 18, 20, 22 and 24 should be specified on the curve.) 3 a Calculate the increase q 1 in height y B , when the ab-scissa x A changes from 24 to 22. Check this result on the curve (Γ) showing the constructions made. b Evaluate, with the aid of the graph showing the useful construction lines, the increase q 2 of y b when x A goes from 22 to 20, then the increase q 2 of y B when x A goes from 20 to 18. c When the crank is operated evenly, can the car be said to climb at a constant speed? Justify your answer. E.2337 The table shows for each line a function and the expression of the derivative. Establish the accuracy of each line. Fonction Image from x Nombre derived in x f (3 x + 2) 6 18 · (3 x + 2) 5 g 4 · (3 − 2 x ) 4 − 32 · (3 − 2 x ) 3 h 1 − 2 x 2 + 3 x + 1 4 x − 3 2 x 2 − 3 x − 1 2 j 5 x 2 + 6 x − 2 5 x + 3 5 x 2 + 6 x − 2 k 2 x − 1 3 − x 2 x − 11 2 · ( x − 3) · 3 − x https://chingmath.fr chapExoCorrec/71 sacados/71 Antilles - 2004 - 7 points - obligatoire OABCIxy chapExoCorrec/2337 sacados/2337