Grade 10 / Table of signs and variations of functions 53 exercises (100% corrected)

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ABCDx012345678f(x0,20,40,60,81Cf x-5-4-3-2-1012y-2-11234 ChingQuizz : 4 exercises available for Quizz assessment : 1. Introduction to the variations of a function E.7064 Consider the function defined on 0 ; 8 by: f ( x ) = − 0.02 · x 2 + 0.2 · x + 0.25 In a mountainous region, a company is studying a road project linking the villages A , B and C located at different altitudes. The function f models the profile of this road project. The variable x represents the horizontal distance, in kilometers, from the village A and f ( x ) represents the associated alti-tude, in kilometers. The graphical representation C f of the function f is given below. For each of the following propositions, only one answer is correct. Which one? Proposition 1 The altitude difference between villages A and B is given by: a f (0) − f (4) b f (4) − f (0) Proposition 2 The altitude difference between the villages C and D is given by: a f (7) − f (8) b f (8) − f (7) . E.378 Consider the function f defined on the interval − 5 ; 2 whose image of a number x is given by the relation: f ( x ) = 1 3 · x 3 + 3 2 · x 2 − 3 2 Consider the plane with the reference frame shown below : Note C f the representative curve of the function f in the above reference frame. 1 Using the calculator, complete the tables of values be-low by entering the values of the images rounded to the nearest tenth : x − 5 − 4.5 − 4 − 3.5 − 3 − 2.5 − 2 − 1.5 f ( x ) x − 1 − 0.5 0 0.5 1 1.5 2 f ( x ) 2 On the interval − 5 ; − 3 , what can be said about the variations of the images by the function f ? On the interval − 3 ; 0 , what can we say about the vari-ations of the images by the function f ? 3 Place the set of points on the curve C f obtained from the previous tables of values. Then plot C f . 4 Simply describe the behavior of the curve C f over the interval − 5 ; − 3 , then on the interval − 3 ; 0 . https://chingmath.fr chapExoCorrec/7064 sacados/7064 ABCDx012345678f(x0,20,40,60,81Cf chapExoCorrec/378 sacados/378 x-5-4-3-2-1012y-2-11234
x-4-3-2-101234y1234Cf x-6-5-4-3-2-10123456y-2-112Cf -51373-1-12xVariationdef E.387 Indication : these questions will be answered with the use of the calculator. 1 Let f be the function defined on R and whose image of a number x is given by the relation: f ( x ) = 2 x + 1 Describe the variations of the function f . 2 Let g be the function defined on R ∗ + by the relation: g ( x ) = 1 x Describe the variations of the function g . 3 Let h be the function defined on − ı ; ı by the relation: h ( x ) = cos( x ) Describe the variations of the function h . Indication : set calculator to radian mode. E.4580 In the plane provided with the ref-erence frame below, consider the curve C f representative of the function f : Describe the variations of the function f on the interval − 4 ; 4 . E.11700 Consider the function f whose curve C f is shown in the O ; I ; J reference below : Which of the boxes below describes the set of values x belong-ing to the definition set of the function f : a − 5 <x< 3.5 b − 5 x< 3.5 c − 5 x 3.5 d − 5 <x 3.5 2. Variation tables E.9478 Consider a function f whose table of variation is given below : Complete the following sentences : the defining set of the function f is D f = ::: the function f is strictly increasing at . . . . . . the function f is strictly decreasing on . . . . . . the function f is . . . . . . on the interval 1 ; 3 The number . . . . . . does not admit an image by f . https://chingmath.fr chapExoCorrec/387 sacados/387 chapExoCorrec/4580 sacados/4580 x-4-3-2-101234y1234Cf chapExoCorrec/11700 sacados/11700 x-6-5-4-3-2-10123456y-2-112Cf chapExoCorrec/9478 sacados/9478 -51373-1-12xVariationdef
x-4-3-2-10123y-20-101020304050 −31244−2039xf(x −3−12441239xf(x −3−2−11244712−2039xf(x −30;61−1;8244−2015539xf(x xVariationdef-4-212-1-321 xVariationdeg-4-312-1-231 xVariationdeh-4-312-1-221 xVariationdej-4-212-1-221 x-4-202y-2246 x-4-202y-2246 x-4-202y-2246 x-4-202y-2246 E.4571 Consider the function f defined on the interval − 3 ; 2 , whose image of a number x is given by the relation: f ( x ) = 3 x 4 + 8 x 3 − 6 x 2 − 24 x − 1 Consider the plane equipped with the orthogonal coordinate system below : Let C f be the representation of the function f in this coordi-nate system. 1 Using a calculator, complete the tables of values below with values rounded to the nearest tenth : x − 3 − 2 ; 8 − 2 ; 4 − 2 − 1 − 0 ; 8 0 f ( x ) x 0 ; 5 0 ; 8 1 1 ; 3 1 ; 5 1 ; 7 2 f ( x ) 2 Plot the curve C f in the coordinate system above. 3 Which of the tables of variations below best represents the curve C f : a b c d E.2722 Below are the tables of variations and graphical representations of four functions f , g , h , j . Associate each table of variations with the corresponding graphical representation : a b c d 1 2 3 4 https://chingmath.fr chapExoCorrec/4571 sacados/4571 x-4-3-2-10123y-20-101020304050 −31244−2039xf(x −3−12441239xf(x −3−2−11244712−2039xf(x −30;61−1;8244−2015539xf(x chapExoCorrec/2722 sacados/2722 xVariationdef-4-212-1-321 xVariationdeg-4-312-1-231 xVariationdeh-4-312-1-221 xVariationdej-4-212-1-221 x-4-202y-2246 x-4-202y-2246 x-4-202y-2246 x-4-202y-2246
xVariationdef-4-212-1-311 xVariationdeg-4-2120213 xVariationdeh-4-2023-2-11 xVariationdej-4-1023-2-11 x-4-202y-2246 x-4-202y-2246 x-4-202y-2246 x-4-202y-2246 x-4-3-2-101234y-2-112Cf x-7-6-5-4-3-2-10123456y-4-3-2-11234CfCf x-4-3-2-101234y481216 −40416016Variationdefet degx x-6-5-4-3-2-10123456y-3-2-112Cg E.356 Below are the variation tables and graphs for four functions : f , g , h , and j . Match each variation table with its corresponding graph : a b c d 1 2 3 4 E.4579 In the plane provided with the ref-erence frame below, consider the curve C f representative of the function f : Draw up the table of variations of the function f . E.360 Consider the plane provided with the orthonormal reference frame below where is given the repre-sentation of the function f defined by the two pieces of curves indicated : 1 Give the set of definition of the function f defined by the representative curve above. 2 a Give, without justification, the images by the func-tion f of the numbers : − 6 ; − 4 ; − 2 ; 0 ; 2.5 b Draw up the table of variations of the function f . 3 Determine the set of antecedents of the number 3 by the function f . E.388 The plane is given the orthogonal reference point shown below : In this frame of reference, draw the representative curves C f and C g of two distinct functions f and g , but admitting the following table of variations : 3. Direction of variation and order E.357 Consider the function g whose repre-sentation is given in the orthonormal frame below : given by the representation below : https://chingmath.fr chapExoCorrec/356 sacados/356 xVariationdef-4-212-1-311 xVariationdeg-4-2120213 xVariationdeh-4-2023-2-11 xVariationdej-4-1023-2-11 x-4-202y-2246 x-4-202y-2246 x-4-202y-2246 x-4-202y-2246 chapExoCorrec/4579 sacados/4579 x-4-3-2-101234y-2-112Cf chapExoCorrec/360 sacados/360 x-7-6-5-4-3-2-10123456y-4-3-2-11234CfCf chapExoCorrec/388 sacados/388 x-4-3-2-101234y481216 −40416016Variationdefet degx chapExoCorrec/357 sacados/357 x-6-5-4-3-2-10123456y-3-2-112Cg
-5-251−2;5xVariationdeg −3035512-3xVariationdef −5−202635−41−2xVariationdef −10−426934−32−1Variationdefx -12-5-92-1036√505-2263-5-30xVariationdef 1 Give the definition set of the function g . 2 The table below represents ˇ schématiquement ı the vari-ations of the function g ; copy and complete this table : 3 a Complete the table of values below : x − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 g ( x ) b What can be said about the two lists of numbers be-low : L 1 = g ( − 5) ; g ( − 4) ; g ( − 3) ; g ( − 2) L 2 = g ( − 2) ; g ( − 1) ; g (0) ; g (1) ; g (2) 4 Complete the following two propositions : Proposition: Let I be an interval of R and a , b two num-bers belonging to I ( a;b ∈ i ) Let f be an increasing function on the interval I . If a <b then f ( a ) : : : f ( b ) . This proposition translates into the sentence : ˇ If a function is increasing then two numbers and their im-ages are ordered . . . . . . ı Let f be a decreasing function on the interval I . If a <b then f ( a ) : : : f ( b ) . This proposition translates into the sentence : ˇ If a function is decreasing then two numbers and their im-ages are ordered . . . . . . ı E.9348 Consider the function f defined on − 3 ; 5 which admits the table of variations below : Carry out comparisons of the pairs of numbers below : a f (0) and f (1) b f (4) and f (5) c f ( − 2) and f ( − 1) d f (1) and f (2) E.9551 Consider a function f defined on − 5 ; 6 and which admits the table of variations below : Compare the numbers below : a f ( − 3) and f ( − 4) b f (3) and f (4) c f ( − 4) and f (4) d f ( − 2) and f (1) E.9586 Let F be the function defined on the interval − 10 ; 9 and whose table of variations is given below : If possible, compare the following pairs of numbers : a f (7) and f (8) b f ( − 9) and f (1) c f ( − 3) and f (3) d f ( − 8) and f ( − 5) E.2704 Consider the function f whose table of variations is given below : Perform, if possible, a comparison of the following numbers : a f ( − 3) et f ( − 2) b f (1) et f (2) c f ( − 5) et f (3) d f (6) et f ( - 4) e f ( - 4.75) et f (7) f f ( - 10) et f ( - 3) g f ( − 6) et f (4) h f (7) et f ( − 2) https://chingmath.fr -5-251−2;5xVariationdeg chapExoCorrec/9348 sacados/9348 −3035512-3xVariationdef chapExoCorrec/9551 sacados/9551 −5−202635−41−2xVariationdef chapExoCorrec/9586 sacados/9586 −10−426934−32−1Variationdefx chapExoCorrec/2704 sacados/2704 -12-5-92-1036√505-2263-5-30xVariationdef
−20√24√3−132-3xVariationdef x-8-7-6-5-4-3-2-101234567y-4-3-2-112345Cf x-11-10-9-8-7-6-5-4-3-2-101y-7-6-5-4-3-2-11Cf E.2725 Consider the function f , whose rep-resentative curve is drawn in a single line. Here is a table of values for the function f : x − 2 − 1 0 0 ; 5 1 2 7 3 4 f ( x ) √ 3 1 − 1 3 1 2 1 2 − 2 − 3 1 Indicate whether the following statements are true, false, or undecidable ; justify your answers : a The function f is decreasing on [0 ; 1] . b The function f is decreasing on [ − 2 ; 0] . c The function f is zero only once. d The maximum value of f is 2 . 2 Suppose that the function f has the following table of variations : With this new information, revisit all of the questions in 1 . 4. Maximum and minimum E.383 The graphical representation of the function f is given in the frame below : 1 Draw up the table of variations of the function f . 2 a What are the coordinates of the highest point of the curve C f ? b Deduce the maximum value taken by the function f on its interval of definition. 3 Give the minimum value taken by the function f and the value of x for which it is reached. E.381 Consider the function f whose graphi-cal representation is given in the orthonormal reference frame below : 1 a Give the definition set of the function f . b Draw up the table of variations of the function f . 2 a Give the coordinates of the highest point of the curve C f . b What is the maximum value reached by the function f ? 3 Give the minimum of the function f on the interval − 10 ; − 5 and the value of x for which this minimum is reached. https://chingmath.fr chapExoCorrec/2725 sacados/2725 −20√24√3−132-3xVariationdef chapExoCorrec/383 sacados/383 x-8-7-6-5-4-3-2-101234567y-4-3-2-112345Cf chapExoCorrec/381 sacados/381 x-11-10-9-8-7-6-5-4-3-2-101y-7-6-5-4-3-2-11Cf
x-6-5-4-3-2-1012y-2-1123CfCf x-1012345678y-112345 x-4-20246y-224Cf x-2024y-224Cg x-8-6-4-202468y-4-224Ch -∞−201∞537-43Variationdefx E.9349 Let f be a function whose graphical representation is given in the orthonormal frame below : 1 Give the definition set of the function f . 2 Draw up the table of variations of the function f . 3 a Give the minimun of the function f on its defining set. b Give the maximun of the function f on the interval − 6 ; − 3 E.382 Here is the graphical representation of a function f . 1 What is the defining set of the function f ? 2 Give the table of variations of the function f ? 3 What is the maximum of the function f on the interval 0 ; 5 2 ? 4 What is the maximum of f on its defining set? 5 What is the minimum of f over [0 ; 7] ? E.359 We consider three functions f , g , h whose representative curves are shown below : 1 For each of these functions, draw up the table of varia-tions from their representative curves. 2 a Give the coordinates of the highest point of the curve C f on the interval − 4 ; 1 ? b Deduce the maximum value reached by the function f on the interval − 4 ; 1 and the value of x for which this value is reached. 3 Give the maximum value taken by the function g ; for what value of x is this maximum reached? 4 a Give the coordinates of the lowest point of the curve C h on the interval − 7 ; 0 ? b Deduce the minimum of the function h on − 7 ; 0 and the value of x for which this value is reached. E.2144 The table of variations of the func-tion f defined at R is shown below : For each of the statements, say whether they are true, false, or undecidable, justifying your answer each time: a 3 admits the number − 2 as its antecedent. b f (1) >f ( − 1) . c f (2) is a positive number. d The minimum of the function f is − 4 . e For x ∈ −∞ ; 0 , we have : f ( x ) 0 f The number 4 admits a single antecedent. 5. Solving inequalities E.6112 Antibiotics are molecules with the ability to kill bacteria or limit their spread. The table below shows the concentration of an antibiotic in-jected into a patient in a single dose, as a function of time. Temps en heure 0.5 1 1.5 2 3 4 5 6 7 8 9 10 Concen tration en mg = 1.6 2 1.9 1.6 1.2 0.9 0.8 0.7 0.6 0.5 0.4 0.4 https://chingmath.fr chapExoCorrec/9349 sacados/9349 x-6-5-4-3-2-1012y-2-1123CfCf chapExoCorrec/382 sacados/382 x-1012345678y-112345 chapExoCorrec/359 sacados/359 x-4-20246y-224Cf x-2024y-224Cg x-8-6-4-202468y-4-224Ch chapExoCorrec/2144 sacados/2144 -∞−201∞537-43Variationdefx chapExoCorrec/6112 sacados/6112
024681012 -7-6-5-4-3-2-123456I-3-2-12345JOCf r513-1 r513-3 r422-1 x01234y1234C1 x01234y1234C2 −2137912350−203xVariationdef These data lead to the modeling of the concentration as a func-tion of time by the function g de-fined on the interval 0 ; 10 by: g ( t ) = 4 · t t 2 + 1 Where t represents the time elapsed, in hours, since the an-tibiotic was injected, g ( t ) repre-sents the concentration in mg = of the antibiotic. The graph opposite represents the data in the table and the representative curve of the function g . By graphical reading give without justification : 1 the variations of the function g on 0 ; 10 ; 2 the maximum antibiotic concentration during the first 10 hours ; 3 the time interval during which the antibiotic concentra-tion in the blood is greater than 1.2 mg = . E.391 The curve below is the representative curve of the function f . 1 Give the definition set of the function f . 2 Draw up the table of variations of the function f . 3 Give, without justification, the images by the function f of the following intervals : a − 2.5 ; 0 b 1 ; 3 c − 6 ; 5 4 For each question, give without justification the set of values of x verifying the following frame : a 3 f ( x ) 4 b − 3 f ( x ) < 1 6. Image of an interval E.4818 Examples: here are some animations illustrating the im-age of an interval by a function. The two graphs below show two parts C 1 and C 2 of the rep-resentative curve of the same function f . 1 a Determine the set of numbers formed by all the ab-scissas of the points on the C 1 part of the curve. b Determine the set of numbers formed by all the ordi-nates of the points on the C 1 part of the curve. c Deduce the image of the interval 2 ; 3.5 by the func-tion f . 2 a Determine the set of numbers formed by all the ab-scissas of the points on the C 2 part of the curve. b Determine the set of numbers formed by all the ordi-nates of the points on the C 2 part of the curve. c Deduce the image of the interval 0 ; 2 by the function f . E.4671 Consider a function f defined on the interval − 2 ; 12 whose table of variations is given below : We call image of an interval I by f the set formed by the image of all numbers in I by the function f . 1 Give, by the function f , the image of the intervals : a 7 ; 12 b 1 ; 3 c − 2 ; 1 2 Give, by the function f , the image of the intervals : a − 2 ; 3 b 3 ; 9 c 1 ; 12 https://chingmath.fr 024681012 chapExoCorrec/391 sacados/391 -7-6-5-4-3-2-123456I-3-2-12345JOCf chapExoCorrec/4818 sacados/4818 r513-1 r513-3 r422-1 x01234y1234C1 x01234y1234C2 chapExoCorrec/4671 sacados/4671 −2137912350−203xVariationdef
x-4-3-2-101234y-3-2-112Cf x-4-3-2-101234y-2-1123Cf x-4-3-2-101234y-2-11Cf −8-4−52015172101215250-305750-2xVariationdef E.4672 In the plane provided with the or-thonormal reference frame below, consider the representation of the function f given below : 1 Give the definition set of the function f . 2 Give the images of the following intervals : a − 4 ; − 2 b − 1 ; 2 c 2 ; 3 3 Give the images of the following intervals : a − 2 ; 0 b 0 ; 4 c − 2 ; 3 E.4798 Consider the function f whose graph-ical representation is given below : Determine the images, by the function f , of each of the inter-vals below : a − 2 ; 0 b − 1 2 ; 3 c − 4 ; 4 E.6693 Consider the function f defined on R whose graphical representation is given in the frame below : Graphically, determine the image of the following intervals by the function f : a 3 2 ; 7 2 b − 1 ; 0 c − 3 ; − 1 4 d − 2 ; 7 2 7. Table of signs E.6563 1 For any x ∈ R , each of the expressions below retains the same sign. Give the sign of each of these expressions, justifying your answer: a x − 1 2 b − 3 x 2 + 1 c 1 + x 2 − 2 − x 2 2 Justify that each of the following statements is false with a counterexample: a The expression − x − 3 is negative for any x ∈ R . b The expression x 2 − 1 is positive for any x ∈ R . c The expression ( x +1)( x +3) is positive for any x ∈ R . E.2781 Consider the function f whose table of variation is given below : 1 From the tables below, indicate the sign table of the func-tion f : a x − 8 − 4 0 17 2 15 f ( x ) + 5 − − 3 + 7 − b x − 8 − 5 2 1 12 15 f ( x ) + 0 − 0 + 0 − https://chingmath.fr chapExoCorrec/4672 sacados/4672 x-4-3-2-101234y-3-2-112Cf chapExoCorrec/4798 sacados/4798 x-4-3-2-101234y-2-1123Cf chapExoCorrec/6693 sacados/6693 x-4-3-2-101234y-2-11Cf chapExoCorrec/6563 sacados/6563 chapExoCorrec/2781 sacados/2781 −8-4−52015172101215250-305750-2xVariationdef
-5-3-102579-2041250-3xVariationdef -3-2-101379-2-40310-2-1xVariationdef −2034710380-201xVariationdef c x − 8 0 15 f ( x ) − 0 + 2 Give the set of solutions of the inequation f ( x ) 0 . E.9352 Consider the function f whose vari-ation table is given below : 1 Give the set of solutions to the inequalities: a f ( x ) < 0 b f ( x ) 0 2 Draw up the sign table for the function f . E.4694 Consider the function f whose table of variations is given below : 1 Determine the images of the following intervals by the function f : a 1 ; 3 b 3 ; 9 2 Compare, if possible and justifying the answer, the fol-lowing pairs of numbers : a f ( − 1.5) ; f (8) b f ( − 0.5) ; f (3.5) c f 9 7 ; f 9 8 d f √ 63 ; f (8) 3 a Give the set of solutions of the inequation : f ( x ) < 0 b Draw up the sign table for the function f . 4 Without justification, give the number of solutions to the equation f ( x )= − 2 . E.2730 Consider the function f de-fined on R admitting the sign table below : x −∞ − 3 5 + ∞ f ( x ) + 0 − 0 + Answer the following statements with ˇ true ı, ˇ faux ı or ˇ we can’t savoir ı : 1 f (2)=6 . 2 The equation f ( x )=0 admits exactly two solutions. 3 The function f is an affine function. 4 The point A (0 ; 5) belongs to the representative curve of the function f . 5 If f (1)= − 4 , then the minimum of the function f on R is − 4 . 8. Table of signs and inequalities E.9850 Consider the function f defined on R admitting the sign table below : x −∞ − 3 5 + ∞ f ( x ) + 0 − 0 + Determine the solutions of the inequation f ( x ) < 0 . 9. Variations and signs E.2732 Consider the function f defined on the interval − 2 ; 10 for which only the table of variations below is given : 1 Describe, in French, the variations of the function f on the interval − 2 ; 10 . 2 a Frame the image of the number 1 by the function f . b Frame the image of the number 6 by the function f . 3 a Give the interval on which the function f is strictly negative. b On what set is the function f strictly positive? https://chingmath.fr chapExoCorrec/9352 sacados/9352 -5-3-102579-2041250-3xVariationdef chapExoCorrec/4694 sacados/4694 -3-2-101379-2-40310-2-1xVariationdef chapExoCorrec/2730 sacados/2730 Extrait du manuel "Ressourcres pour la classe de seconde" - dgesco 2009 chapExoCorrec/9850 sacados/9850 chapExoCorrec/2732 sacados/2732 −2034710380-201xVariationdef
x-6-5-4-3-2-10123y-2-1123Cf x-6-5-4-3-2-1012y-2-1123CfCf −2137912350−203xVariationdef -4-3-2-12345I-3-2-123JO −7-2−1202325√31730-2034103xVariationdef E.6565 Consider a function f whose repre-sentative curve C f is given in the orthonormal frame below : 1 Give the definition set of the function f . 2 Draw up the table of variations of the function f . 3 Draw up the table of signs of the function f . E.4695 Let f be a function whose graphical representation is given in the orthonormal frame below : 1 Give the definition set of the function f . 2 Draw up the table of variations of the function f . 3 Draw up the table of signs of the function f on its set of definition. E.8311 Consider a function f defined on the interval − 2 ; 12 whose table of variations is given below : 1 Compare, if possible, the following numbers : a f ( − 1) and f (8) b f (2) and f (5) c f (0) and f (10) 2 Give, by the function f , the image of the intervals : a 1 ; 3 b 1 ; 12 3 Draw up the sign table for the function f . E.2146 Consider a function f verifying each of the following assertions : The function is defined on − 3.5 ; 4 . It is strictly increasing on − 3.5 ; − 1 and strictly decreas-ing on − 1 ; 4 ; The number 2 has a unique antecedent ; The equation f ( x ) 0 admits for set of solutions the in-terval − 2 ; 1 . 1 Give the antecedents of 0 by the function f . 2 What is the maximum of the function f ? For what value is it reached? 3 Draw, freehand, a curve that can represent the function f in the ( O ; I ; J ) orthonormal frame below : 10. Study of functions and tables of variations E.2727 Consider the function f , defined on the interval − 7 ; 31 for which only the table of variations below is given : https://chingmath.fr chapExoCorrec/6565 sacados/6565 x-6-5-4-3-2-10123y-2-1123Cf chapExoCorrec/4695 sacados/4695 x-6-5-4-3-2-1012y-2-1123CfCf chapExoCorrec/8311 sacados/8311 −2137912350−203xVariationdef chapExoCorrec/2146 sacados/2146 -4-3-2-12345I-3-2-123JO chapExoCorrec/2727 sacados/2727 −7-2−1202325√31730-2034103xVariationdef
xVariationdef-8-2016537-43 xVariationdef−10-206537-4 1 a Give, if possible, the set of antecedents of the num-ber 0 by the function f . b Give the sign table for the function f . 2 Solve the inequation : f ( x ) 3 . 3 Give the maximum and minimum of the function f and the values for which they are reached. E.358 Consider a function f defined on − 8 ; 6 whose table of variations has been given below : Say, without justification, whether the statements below are true or false : a − 2 is an antecedent of the number 3 b f (1) >f ( − 1) c f (1) is a number positif d Pour x ∈ ]0 ; 1[ , we have f ( x ) 0 e Le minimum of the function f is − 4. E.1804 Consider a function f defined on the interval − 10 ; 6 whose table of variations is shown below : For each of the statements below, say whether they are true, false or undecidable, each time justifying your thinking : a f ( − 10) <f ( − 1) b Le minimum of f is reached in − 2 c f (1) <f 2 d f (1) is a number positif 11. Algebraic study E.370 Let f be the function defined on − 3 ; 7 whose image of a number x of this interval is given by the formula : f ( x ) = − 1 4 x 2 + x + 3 . 1 Determine the images, by the function f , of the numbers : a 2 b 3 c 2 + 1 2 a Draw the representative curve of the function f us-ing your calculator. b Using your calculator, determine the coordinates of the highest point of C f . 3 a Establish the relationship below : f ( x ) = − 1 2 · x − 1 2 + 4 b Deduce that for any number x belonging to the inter-val − 3 ; 7 , we have : f ( x ) 4 4 Confirm the observation made in question 2 b . E.4697 Consider the function f defined on R by: f ( x )= 2 − x 2 x 2 +1 1 Using the calculator, give the extremums of this function. 2 a Establish equality: f ( x )= 3 x 2 +1 − 1 b Justify the inequality: 3 x 2 +1 3 . c Find the result of question 1 . E.4698 Consider the function f defined on R by: f ( x ) = − x 2 + 4 x − 2 x 2 − 4 x + 5 1 Using the calculator, give the extremums of this function. 2 a Establish equality: f ( x )= 3 1+( x − 2) 2 − 1 b Justify the inequality: 3 1+( x − 2) 2 3 . c Find the result of question 1 . 12. Variation and sign table https://chingmath.fr chapExoCorrec/358 sacados/358 xVariationdef-8-2016537-43 chapExoCorrec/1804 sacados/1804 xVariationdef−10-206537-4 chapExoCorrec/370 sacados/370 chapExoCorrec/4697 sacados/4697 chapExoCorrec/4698 sacados/4698
−∞1∞−∞50−∞xVariationdefa −∞-1∞−∞55−∞xVariationdefb −∞2∞−∞-13−∞xVariationdefc −∞4∞−∞34−∞xVariationdefs 03591538424xVariationdef -3-2-101379-2-40310-2-1xVariationdef -5-3-102579-2041250-3xVariationdef E.11049 Consider the function f defined by: f ( x ) = − 18 x 2 + 36 x + 32 1 a Establish equality: f ( x ) = 6 x + 4 8 − 3 x b Draw up the sign table for the function f . 2 Which of the following tables of variations is that of the function f . 13. Variation and prime numbers E.11050 Consider the function f , whose ta-ble of variations is known : Of the intervals 0 ; 3 and 5 ; 9 , which one has the image by the function f that contains the most prime integers? 14. Variation and probability E.11051 Consider the function f whose vari-ation table is given below : Choose a random integer in the interval − 3 ; 9 . What is the probability that the image of this integer, by the function f , is strictly negative? 15. Share E.4678 Consider the function f whose table of variations is given below : 1 Determine the images of the following intervals by the function f : a − 5 ; − 3 b − 1 ; 0 c 2 ; 9 2 Compare, if possible, the following pairs of numbers : a f ( − 4) ; f ( − 2) b f (6) ; f (8) c f (1) ; f (8) d f (3) ; f (4) e f − 2 3 ; f − 1 2 f f ( − 2) ; f (3) 3 Draw up the sign table for the function f . 4 Without justification, give the number of solutions to the equation f ( x )=4 . 16. Unclassified exercises https://chingmath.fr chapExoCorrec/11049 sacados/11049 −∞1∞−∞50−∞xVariationdefa −∞-1∞−∞55−∞xVariationdefb −∞2∞−∞-13−∞xVariationdefc −∞4∞−∞34−∞xVariationdefs chapExoCorrec/11050 sacados/11050 03591538424xVariationdef chapExoCorrec/11051 sacados/11051 -3-2-101379-2-40310-2-1xVariationdef chapExoCorrec/4678 sacados/4678 -5-3-102579-2041250-3xVariationdef
xVariationdef−3025∞−22−1 E.2196 Consider the function f whose table of variations is as follows : 1 Say whether the following assertions are true, false or undecidable. In each case, justify your assertion : a D f = [ − 2 ; + ∞ [ b The number 2, through the function f , admits only one antecedent. c f is bounded on its defining set. d The image of 4 is a negative number. 2 The following information is given about the function f : the image of − 1 (resp. 3) by the function f is 3 (resp. 0) . Give, without justification, the image of the intervals be-low by the function f : a [0 ; 5] b [ − 1 ; 2] c ] − 3 ; 3[ https://chingmath.fr chapExoCorrec/2196 sacados/2196 xVariationdef−3025∞−22−1