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Trazado el gráfico de la función/ abs(15-2*x-x^2)+abs(x^2-x-2)-13+3*x

Gráfico de la función y = abs(15-2*x-x^2)+abs(x^2-x-2)-13+3*x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       |            2|   | 2        |           
f(x) = |15 - 2*x - x | + |x  - x - 2| - 13 + 3*x
f(x)=3x+((∣−x2+(15−2x)∣+∣(x2−x)−2∣)−13)f{\left(x \right)} = 3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right) 
f = 3*x + |-x^2 + 15 - 2*x| + |x^2 - x - 2| - 13
Gráfico de la función
02468-8-6-4-2-1010-250250
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
3x+((∣−x2+(15−2x)∣+∣(x2−x)−2∣)−13)=03 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−5x_{1} = -5
x2=3x_{2} = 3
x3=−2x_{3} = -2
x4=−4x_{4} = -4
x5=2x_{5} = 2
x6=−2.66666666666667x_{6} = -2.66666666666667
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en |15 - 2*x - x^2| + |x^2 - x - 2| - 13 + 3*x.
0⋅3+(−13+(∣−2+(02−0)∣+∣−02+(15−0)∣))0 \cdot 3 + \left(-13 + \left(\left|{-2 + \left(0^{2} - 0\right)}\right| + \left|{- 0^{2} + \left(15 - 0\right)}\right|\right)\right)
Resultado:
f(0)=4f{\left(0 \right)} = 4
Punto:
(0, 4)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
(−2x−2)sign⁡(−x2−2x+15)+(2x−1)sign⁡(x2−x−2)+3=0\left(- 2 x - 2\right) \operatorname{sign}{\left(- x^{2} - 2 x + 15 \right)} + \left(2 x - 1\right) \operatorname{sign}{\left(x^{2} - x - 2 \right)} + 3 = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1.75872093023256x_{1} = -1.75872093023256
x2=−1.80357142857143x_{2} = -1.80357142857143
x3=−1.75716560509554x_{3} = -1.75716560509554
x4=−1.69078947368421x_{4} = -1.69078947368421
x5=−1.76032110091743x_{5} = -1.76032110091743
x6=−1.75582901554404x_{6} = -1.75582901554404
x7=−1.76973684210526x_{7} = -1.76973684210526
x8=−1.74326347305389x_{8} = -1.74326347305389
x9=−1.76113861386139x_{9} = -1.76113861386139
x10=0.5x_{10} = 0.5
x11=−1.64772727272727x_{11} = -1.64772727272727
x12=−1.70108695652174x_{12} = -1.70108695652174
x13=−1.72383720930233x_{13} = -1.72383720930233
x14=−1.72794117647059x_{14} = -1.72794117647059
x15=−1.83653846153846x_{15} = -1.83653846153846
x16=−1.73907766990291x_{16} = -1.73907766990291
x17=2.25x_{17} = 2.25
x18=−1.74423076923077x_{18} = -1.74423076923077
x19=−1.74021739130435x_{19} = -1.74021739130435
x20=−1.77122641509434x_{20} = -1.77122641509434
x21=−1.73214285714286x_{21} = -1.73214285714286
x22=−1.75735294117647x_{22} = -1.75735294117647
x23=−1.76388888888889x_{23} = -1.76388888888889
x24=−1.72954545454545x_{24} = -1.72954545454545
x25=−1.74054621848739x_{25} = -1.74054621848739
x26=−1.76541095890411x_{26} = -1.76541095890411
x27=−1.74085365853659x_{27} = -1.74085365853659
x28=−1.72606382978723x_{28} = -1.72606382978723
x29=−1.73093220338983x_{29} = -1.73093220338983
x30=−1.73763736263736x_{30} = -1.73763736263736
x31=−1.7626404494382x_{31} = -1.7626404494382
x32=−1.81617647058824x_{32} = -1.81617647058824
x33=−1.71785714285714x_{33} = -1.71785714285714
x34=−1.73863636363636x_{34} = -1.73863636363636
x35=−1.77743902439024x_{35} = -1.77743902439024
x36=−1.74371508379888x_{36} = -1.74371508379888
x37=−1.76730769230769x_{37} = -1.76730769230769
x38=−1.72115384615385x_{38} = -1.72115384615385
x39=−1.75775862068966x_{39} = -1.75775862068966
x40=−1.75845864661654x_{40} = -1.75845864661654
x41=−1.74342105263158x_{41} = -1.74342105263158
x42=−1.74292452830189x_{42} = -1.74292452830189
x43=−1.76323529411765x_{43} = -1.76323529411765
x44=−1.795x_{44} = -1.795
x45=−1.75595238095238x_{45} = -1.75595238095238
x46=−2x_{46} = -2
x47=−1.76209677419355x_{47} = -1.76209677419355
x48=−1.74254966887417x_{48} = -1.74254966887417
x49=−1.775x_{49} = -1.775
x50=−1.76159793814433x_{50} = -1.76159793814433
x51=−1.75608108108108x_{51} = -1.75608108108108
x52=−1.74274193548387x_{52} = -1.74274193548387
x53=−1.70833333333333x_{53} = -1.70833333333333
x54=−1.7430981595092x_{54} = -1.7430981595092
x55=−1.78409090909091x_{55} = -1.78409090909091
x56=−1.73948598130841x_{56} = -1.73948598130841
x57=−1.73644578313253x_{57} = -1.73644578313253
x58=−1.74234693877551x_{58} = -1.74234693877551
x59=−1.75681818181818x_{59} = -1.75681818181818
x60=−1.75650289017341x_{60} = -1.75650289017341
x61=−1.73986486486486x_{61} = -1.73986486486486
x62=−1.875x_{62} = -1.875
x63=−1.73815789473684x_{63} = -1.73815789473684
x64=−1.74190647482014x_{64} = -1.74190647482014
x65=−1.75698757763975x_{65} = -1.75698757763975
x66=−1.75755033557047x_{66} = -1.75755033557047
x67=−1.75571065989848x_{67} = -1.75571065989848
x68=−1.73320895522388x_{68} = -1.73320895522388
x69=−1.74398395721925x_{69} = -1.74398395721925
x70=−1.675x_{70} = -1.675
x71=−1.73706896551724x_{71} = -1.73706896551724
x72=−1.74141221374046x_{72} = -1.74141221374046
x73=−1.74410994764398x_{73} = -1.74410994764398
x74=−1.74114173228346x_{74} = -1.74114173228346
x75=−1.74357142857143x_{75} = -1.74357142857143
x76=−1.75665680473373x_{76} = -1.75665680473373
x77=−1.73575949367089x_{77} = -1.73575949367089
x78=−1.73415492957746x_{78} = -1.73415492957746
x79=−1.76461038961039x_{79} = -1.76461038961039
x80=−1.71370967741935x_{80} = -1.71370967741935
x81=−1.75797872340426x_{81} = -1.75797872340426
x82=−1.75929752066116x_{82} = -1.75929752066116
x83=−1.74434673366834x_{83} = -1.74434673366834
x84=−1.75621546961326x_{84} = -1.75621546961326
x85=−1.75995575221239x_{85} = -1.75995575221239
x86=−1.76630434782609x_{86} = -1.76630434782609
x87=−1.75961538461538x_{87} = -1.75961538461538
x88=−1.76071428571429x_{88} = -1.76071428571429
x89=−1.75559701492537x_{89} = -1.75559701492537
x90=−1.75821167883212x_{90} = -1.75821167883212
x91=−1.759x_{91} = -1.759
x92=−1.78879310344828x_{92} = -1.78879310344828
x93=−1.74385245901639x_{93} = -1.74385245901639
x94=−1.77295918367347x_{94} = -1.77295918367347
x95=−1.74213286713287x_{95} = -1.74213286713287
x96=−1.75635593220339x_{96} = -1.75635593220339
x97=−1.735x_{97} = -1.735
x98=−1.76844262295082x_{98} = -1.76844262295082
x99=−1.78040540540541x_{99} = -1.78040540540541
x100=−4x_{100} = -4
x101=−1.74166666666667x_{101} = -1.74166666666667
Signos de extremos en los puntos:
(-1.7587209302325582, 8.88178419700125e-16)

(-1.8035714285714286, 2.66453525910038e-15)

(-1.7571656050955413, 8.88178419700125e-16)

(-1.6907894736842106, -1.77635683940025e-15)

(-1.760321100917431, -2.66453525910038e-15)

(-1.7558290155440415, -1.77635683940025e-15)

(-1.769736842105263, 1.77635683940025e-15)

(-1.7432634730538923, 0)

(-1.761138613861386, 0)

(0.5, 4.5)

(-1.6477272727272727, 1.77635683940025e-15)

(-1.701086956521739, -1.77635683940025e-15)

(-1.7238372093023255, 3.5527136788005e-15)

(-1.7279411764705883, 3.5527136788005e-15)

(-1.8365384615384615, 2.66453525910038e-15)

(-1.7390776699029127, -2.66453525910038e-15)

(2.25, 0)

(-1.7442307692307693, 3.5527136788005e-15)

(-1.7402173913043477, 1.77635683940025e-15)

(-1.7712264150943395, 8.88178419700125e-16)

(-1.7321428571428572, 1.77635683940025e-15)

(-1.7573529411764706, 1.77635683940025e-15)

(-1.7638888888888888, 1.77635683940025e-15)

(-1.7295454545454545, 2.66453525910038e-15)

(-1.740546218487395, 1.77635683940025e-15)

(-1.7654109589041096, 1.77635683940025e-15)

(-1.7408536585365855, 0)

(-1.726063829787234, 0)

(-1.7309322033898304, 0)

(-1.7376373626373627, 0)

(-1.7626404494382022, -1.77635683940025e-15)

(-1.8161764705882353, 8.88178419700125e-16)

(-1.7178571428571427, 0)

(-1.7386363636363635, -8.88178419700125e-16)

(-1.7774390243902438, 0)

(-1.7437150837988826, -8.88178419700125e-16)

(-1.7673076923076922, -1.77635683940025e-15)

(-1.7211538461538463, 1.77635683940025e-15)

(-1.7577586206896552, -1.77635683940025e-15)

(-1.7584586466165413, 0)

(-1.743421052631579, -8.88178419700125e-16)

(-1.7429245283018868, 0)

(-1.763235294117647, 0)

(-1.795, -1.77635683940025e-15)

(-1.755952380952381, 0)

(-2, 0)

(-1.7620967741935485, 1.77635683940025e-15)

(-1.7425496688741722, 8.88178419700125e-16)

(-1.775, 3.5527136788005e-15)

(-1.7615979381443299, 1.77635683940025e-15)

(-1.7560810810810812, 8.88178419700125e-16)

(-1.742741935483871, -1.77635683940025e-15)

(-1.7083333333333333, 0)

(-1.7430981595092025, -1.77635683940025e-15)

(-1.7840909090909092, -8.88178419700125e-16)

(-1.7394859813084111, 0)

(-1.7364457831325302, 2.66453525910038e-15)

(-1.7423469387755102, -1.77635683940025e-15)

(-1.7568181818181818, 2.66453525910038e-15)

(-1.7565028901734103, 0)

(-1.739864864864865, -1.77635683940025e-15)

(-1.875, 0)

(-1.7381578947368421, 1.77635683940025e-15)

(-1.7419064748201438, 1.77635683940025e-15)

(-1.7569875776397517, 0)

(-1.7575503355704698, 0)

(-1.755710659898477, 0)

(-1.7332089552238805, -1.77635683940025e-15)

(-1.7439839572192513, 2.66453525910038e-15)

(-1.675, 1.77635683940025e-15)

(-1.7370689655172413, 1.77635683940025e-15)

(-1.741412213740458, 0)

(-1.744109947643979, 2.66453525910038e-15)

(-1.7411417322834646, -8.88178419700125e-16)

(-1.7435714285714285, -8.88178419700125e-16)

(-1.7566568047337279, 2.66453525910038e-15)

(-1.735759493670886, 0)

(-1.7341549295774648, -1.77635683940025e-15)

(-1.7646103896103895, 0)

(-1.7137096774193548, 0)

(-1.7579787234042554, 8.88178419700125e-16)

(-1.759297520661157, 0)

(-1.7443467336683418, 0)

(-1.7562154696132597, -8.88178419700125e-16)

(-1.7599557522123894, 0)

(-1.766304347826087, -2.66453525910038e-15)

(-1.7596153846153846, 0)

(-1.7607142857142857, 1.77635683940025e-15)

(-1.7555970149253732, 0)

(-1.7582116788321167, -8.88178419700125e-16)

(-1.759, 1.77635683940025e-15)

(-1.7887931034482758, -1.77635683940025e-15)

(-1.7438524590163935, 8.88178419700125e-16)

(-1.7729591836734695, -1.77635683940025e-15)

(-1.742132867132867, -1.77635683940025e-15)

(-1.75635593220339, 0)

(-1.735, -1.77635683940025e-15)

(-1.7684426229508197, 0)

(-1.7804054054054055, 8.88178419700125e-16)

(-4, 0)

(-1.7416666666666667, 1.77635683940025e-15)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=−1.76032110091743x_{1} = -1.76032110091743
x2=−1.75582901554404x_{2} = -1.75582901554404
x3=−1.70108695652174x_{3} = -1.70108695652174
x4=−1.73863636363636x_{4} = -1.73863636363636
x5=−1.74371508379888x_{5} = -1.74371508379888
x6=−1.74342105263158x_{6} = -1.74342105263158
x7=−1.74274193548387x_{7} = -1.74274193548387
x8=−1.7430981595092x_{8} = -1.7430981595092
x9=−1.75755033557047x_{9} = -1.75755033557047
x10=−1.75621546961326x_{10} = -1.75621546961326
x11=−1.76630434782609x_{11} = -1.76630434782609
x12=−1.74213286713287x_{12} = -1.74213286713287
Puntos máximos de la función:
x12=−1.76973684210526x_{12} = -1.76973684210526
x12=0.5x_{12} = 0.5
x12=−1.72383720930233x_{12} = -1.72383720930233
x12=−1.72794117647059x_{12} = -1.72794117647059
x12=−1.74423076923077x_{12} = -1.74423076923077
x12=−1.77122641509434x_{12} = -1.77122641509434
x12=−1.73214285714286x_{12} = -1.73214285714286
x12=−1.75735294117647x_{12} = -1.75735294117647
x12=−1.76388888888889x_{12} = -1.76388888888889
x12=−1.72954545454545x_{12} = -1.72954545454545
x12=−1.74054621848739x_{12} = -1.74054621848739
x12=−1.81617647058824x_{12} = -1.81617647058824
x12=−1.72115384615385x_{12} = -1.72115384615385
x12=−1.75595238095238x_{12} = -1.75595238095238
x12=−1.74254966887417x_{12} = -1.74254966887417
x12=−1.775x_{12} = -1.775
x12=−1.76159793814433x_{12} = -1.76159793814433
x12=−1.75608108108108x_{12} = -1.75608108108108
x12=−1.73644578313253x_{12} = -1.73644578313253
x12=−1.74190647482014x_{12} = -1.74190647482014
x12=−1.675x_{12} = -1.675
x12=−1.73706896551724x_{12} = -1.73706896551724
x12=−1.76071428571429x_{12} = -1.76071428571429
x12=−1.759x_{12} = -1.759
x12=−1.74166666666667x_{12} = -1.74166666666667
Decrece en los intervalos
[−1.70108695652174,∞)\left[-1.70108695652174, \infty\right)
Crece en los intervalos
(−∞,−1.76630434782609]\left(-\infty, -1.76630434782609\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
2(4(x+1)2δ(x2+2x−15)+(2x−1)2δ(−x2+x+2)−sign⁡(−x2+x+2)+sign⁡(x2+2x−15))=02 \left(4 \left(x + 1\right)^{2} \delta\left(x^{2} + 2 x - 15\right) + \left(2 x - 1\right)^{2} \delta\left(- x^{2} + x + 2\right) - \operatorname{sign}{\left(- x^{2} + x + 2 \right)} + \operatorname{sign}{\left(x^{2} + 2 x - 15 \right)}\right) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(3x+((∣−x2+(15−2x)∣+∣(x2−x)−2∣)−13))=∞\lim_{x \to -\infty}\left(3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(3x+((∣−x2+(15−2x)∣+∣(x2−x)−2∣)−13))=∞\lim_{x \to \infty}\left(3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función |15 - 2*x - x^2| + |x^2 - x - 2| - 13 + 3*x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(3x+((∣−x2+(15−2x)∣+∣(x2−x)−2∣)−13)x)=−∞\lim_{x \to -\infty}\left(\frac{3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(3x+((∣−x2+(15−2x)∣+∣(x2−x)−2∣)−13)x)=∞\lim_{x \to \infty}\left(\frac{3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)}{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
3x+((∣−x2+(15−2x)∣+∣(x2−x)−2∣)−13)=−3x+∣−x2+2x+15∣+∣x2+x−2∣−133 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right) = - 3 x + \left|{- x^{2} + 2 x + 15}\right| + \left|{x^{2} + x - 2}\right| - 13
- No
3x+((∣−x2+(15−2x)∣+∣(x2−x)−2∣)−13)=3x−∣−x2+2x+15∣−∣x2+x−2∣+133 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right) = 3 x - \left|{- x^{2} + 2 x + 15}\right| - \left|{x^{2} + x - 2}\right| + 13
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор