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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
$$f{\left(x \right)} = 3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)$$
f = 3*x + |-x^2 + 15 - 2*x| + |x^2 - x - 2| - 13
Gráfico de la función
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:
$$3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
$$x_{1} = -5$$
$$x_{2} = 3$$
$$x_{3} = -2$$
$$x_{4} = -4$$
$$x_{5} = 2$$
$$x_{6} = -2.66666666666667$$
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.
$$0 \cdot 3 + \left(-13 + \left(\left|{-2 + \left(0^{2} - 0\right)}\right| + \left|{- 0^{2} + \left(15 - 0\right)}\right|\right)\right)$$
Resultado:
$$f{\left(0 \right)} = 4$$
Punto:
(0, 4)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\left(- 2 x - 2\right) \operatorname{sign}{\left(- x^{2} - 2 x + 15 \right)} + \left(2 x - 1\right) \operatorname{sign}{\left(x^{2} - x - 2 \right)} + 3 = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -1.75872093023256$$
$$x_{2} = -1.80357142857143$$
$$x_{3} = -1.75716560509554$$
$$x_{4} = -1.69078947368421$$
$$x_{5} = -1.76032110091743$$
$$x_{6} = -1.75582901554404$$
$$x_{7} = -1.76973684210526$$
$$x_{8} = -1.74326347305389$$
$$x_{9} = -1.76113861386139$$
$$x_{10} = 0.5$$
$$x_{11} = -1.64772727272727$$
$$x_{12} = -1.70108695652174$$
$$x_{13} = -1.72383720930233$$
$$x_{14} = -1.72794117647059$$
$$x_{15} = -1.83653846153846$$
$$x_{16} = -1.73907766990291$$
$$x_{17} = 2.25$$
$$x_{18} = -1.74423076923077$$
$$x_{19} = -1.74021739130435$$
$$x_{20} = -1.77122641509434$$
$$x_{21} = -1.73214285714286$$
$$x_{22} = -1.75735294117647$$
$$x_{23} = -1.76388888888889$$
$$x_{24} = -1.72954545454545$$
$$x_{25} = -1.74054621848739$$
$$x_{26} = -1.76541095890411$$
$$x_{27} = -1.74085365853659$$
$$x_{28} = -1.72606382978723$$
$$x_{29} = -1.73093220338983$$
$$x_{30} = -1.73763736263736$$
$$x_{31} = -1.7626404494382$$
$$x_{32} = -1.81617647058824$$
$$x_{33} = -1.71785714285714$$
$$x_{34} = -1.73863636363636$$
$$x_{35} = -1.77743902439024$$
$$x_{36} = -1.74371508379888$$
$$x_{37} = -1.76730769230769$$
$$x_{38} = -1.72115384615385$$
$$x_{39} = -1.75775862068966$$
$$x_{40} = -1.75845864661654$$
$$x_{41} = -1.74342105263158$$
$$x_{42} = -1.74292452830189$$
$$x_{43} = -1.76323529411765$$
$$x_{44} = -1.795$$
$$x_{45} = -1.75595238095238$$
$$x_{46} = -2$$
$$x_{47} = -1.76209677419355$$
$$x_{48} = -1.74254966887417$$
$$x_{49} = -1.775$$
$$x_{50} = -1.76159793814433$$
$$x_{51} = -1.75608108108108$$
$$x_{52} = -1.74274193548387$$
$$x_{53} = -1.70833333333333$$
$$x_{54} = -1.7430981595092$$
$$x_{55} = -1.78409090909091$$
$$x_{56} = -1.73948598130841$$
$$x_{57} = -1.73644578313253$$
$$x_{58} = -1.74234693877551$$
$$x_{59} = -1.75681818181818$$
$$x_{60} = -1.75650289017341$$
$$x_{61} = -1.73986486486486$$
$$x_{62} = -1.875$$
$$x_{63} = -1.73815789473684$$
$$x_{64} = -1.74190647482014$$
$$x_{65} = -1.75698757763975$$
$$x_{66} = -1.75755033557047$$
$$x_{67} = -1.75571065989848$$
$$x_{68} = -1.73320895522388$$
$$x_{69} = -1.74398395721925$$
$$x_{70} = -1.675$$
$$x_{71} = -1.73706896551724$$
$$x_{72} = -1.74141221374046$$
$$x_{73} = -1.74410994764398$$
$$x_{74} = -1.74114173228346$$
$$x_{75} = -1.74357142857143$$
$$x_{76} = -1.75665680473373$$
$$x_{77} = -1.73575949367089$$
$$x_{78} = -1.73415492957746$$
$$x_{79} = -1.76461038961039$$
$$x_{80} = -1.71370967741935$$
$$x_{81} = -1.75797872340426$$
$$x_{82} = -1.75929752066116$$
$$x_{83} = -1.74434673366834$$
$$x_{84} = -1.75621546961326$$
$$x_{85} = -1.75995575221239$$
$$x_{86} = -1.76630434782609$$
$$x_{87} = -1.75961538461538$$
$$x_{88} = -1.76071428571429$$
$$x_{89} = -1.75559701492537$$
$$x_{90} = -1.75821167883212$$
$$x_{91} = -1.759$$
$$x_{92} = -1.78879310344828$$
$$x_{93} = -1.74385245901639$$
$$x_{94} = -1.77295918367347$$
$$x_{95} = -1.74213286713287$$
$$x_{96} = -1.75635593220339$$
$$x_{97} = -1.735$$
$$x_{98} = -1.76844262295082$$
$$x_{99} = -1.78040540540541$$
$$x_{100} = -4$$
$$x_{101} = -1.74166666666667$$
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:
$$x_{1} = -1.76032110091743$$
$$x_{2} = -1.75582901554404$$
$$x_{3} = -1.70108695652174$$
$$x_{4} = -1.73863636363636$$
$$x_{5} = -1.74371508379888$$
$$x_{6} = -1.74342105263158$$
$$x_{7} = -1.74274193548387$$
$$x_{8} = -1.7430981595092$$
$$x_{9} = -1.75755033557047$$
$$x_{10} = -1.75621546961326$$
$$x_{11} = -1.76630434782609$$
$$x_{12} = -1.74213286713287$$
Puntos máximos de la función:
$$x_{12} = -1.76973684210526$$
$$x_{12} = 0.5$$
$$x_{12} = -1.72383720930233$$
$$x_{12} = -1.72794117647059$$
$$x_{12} = -1.74423076923077$$
$$x_{12} = -1.77122641509434$$
$$x_{12} = -1.73214285714286$$
$$x_{12} = -1.75735294117647$$
$$x_{12} = -1.76388888888889$$
$$x_{12} = -1.72954545454545$$
$$x_{12} = -1.74054621848739$$
$$x_{12} = -1.81617647058824$$
$$x_{12} = -1.72115384615385$$
$$x_{12} = -1.75595238095238$$
$$x_{12} = -1.74254966887417$$
$$x_{12} = -1.775$$
$$x_{12} = -1.76159793814433$$
$$x_{12} = -1.75608108108108$$
$$x_{12} = -1.73644578313253$$
$$x_{12} = -1.74190647482014$$
$$x_{12} = -1.675$$
$$x_{12} = -1.73706896551724$$
$$x_{12} = -1.76071428571429$$
$$x_{12} = -1.759$$
$$x_{12} = -1.74166666666667$$
Decrece en los intervalos
$$\left[-1.70108695652174, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -1.76630434782609\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$2 \left(4 \left(x + 1\right)^{2} \delta\left(x^{2} + 2 x - 15\right) + \left(2 x - 1\right)^{2} \delta\left(- x^{2} + x + 2\right) - \operatorname{sign}{\left(- x^{2} + x + 2 \right)} + \operatorname{sign}{\left(x^{2} + 2 x - 15 \right)}\right) = 0$$
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
$$\lim_{x \to -\infty}\left(3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
$$\lim_{x \to \infty}\left(3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)\right) = \infty$$
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
$$\lim_{x \to -\infty}\left(\frac{3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)}{x}\right) = -\infty$$
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
$$\lim_{x \to \infty}\left(\frac{3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right)}{x}\right) = \infty$$
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:
$$3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right) = - 3 x + \left|{- x^{2} + 2 x + 15}\right| + \left|{x^{2} + x - 2}\right| - 13$$
- No
$$3 x + \left(\left(\left|{- x^{2} + \left(15 - 2 x\right)}\right| + \left|{\left(x^{2} - x\right) - 2}\right|\right) - 13\right) = 3 x - \left|{- x^{2} + 2 x + 15}\right| - \left|{x^{2} + x - 2}\right| + 13$$
- No
es decir, función
no es
par ni impar