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Trazado el gráfico de la función/ (1/2)(abs(x/4+4/x)+x/4+4/x)

Gráfico de la función y = (1/2)(abs(x/4+4/x)+x/4+4/x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       |x   4|   x   4
       |- + -| + - + -
       |4   x|   4   x
f(x) = ---------------
              2
f(x)=(x4+∣x4+4x∣)+4x2f{\left(x \right)} = \frac{\left(\frac{x}{4} + \left|{\frac{x}{4} + \frac{4}{x}}\right|\right) + \frac{4}{x}}{2} 
f = (x/4 + |x/4 + 4/x| + 4/x)/2
Gráfico de la función
02468-8-6-4-2-1010-100100
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
(x4+∣x4+4x∣)+4x2=0\frac{\left(\frac{x}{4} + \left|{\frac{x}{4} + \frac{4}{x}}\right|\right) + \frac{4}{x}}{2} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−0.573069559668156x_{1} = -0.573069559668156
x2=−68.3184666390594x_{2} = -68.3184666390594
x3=−0.471351351351351x_{3} = -0.471351351351351
x4=−0.848695652173913x_{4} = -0.848695652173913
x5=−0.950805398345668x_{5} = -0.950805398345668
x6=−0.410677618069815x_{6} = -0.410677618069815
x7=−93.7607979893689x_{7} = -93.7607979893689
x8=−0.400374765771393x_{8} = -0.400374765771393
x9=−1.01181102362205x_{9} = -1.01181102362205
x10=−59.7809343434343x_{10} = -59.7809343434343
x11=−28.279689608637x_{11} = -28.279689608637
x12=−0.355763893172146x_{12} = -0.355763893172146
x13=−69.0925583214749x_{13} = -69.0925583214749
x14=−0.805520702634881x_{14} = -0.805520702634881
x15=−25.9381975446429x_{15} = -25.9381975446429
x16=−47.8118359067269x_{16} = -47.8118359067269
x17=−0.320111899290638x_{17} = -0.320111899290638
x18=−45.4336930003498x_{18} = -45.4336930003498
x19=−1.856x_{19} = -1.856
x20=−41.842342179092x_{20} = -41.842342179092
x21=−2.46321525885559x_{21} = -2.46321525885559
x22=−57.7416002935287x_{22} = -57.7416002935287
x23=−0.381249114856253x_{23} = -0.381249114856253
x24=−71.9570314031057x_{24} = -71.9570314031057
x25=−0.445044350173544x_{25} = -0.445044350173544
x26=−49.8667344929056x_{26} = -49.8667344929056
x27=−0.642499499299019x_{27} = -0.642499499299019
x28=−77.4950279791414x_{28} = -77.4950279791414
x29=−20.732313239645x_{29} = -20.732313239645
x30=−0.5009765625x_{30} = -0.5009765625
x31=−0.37235695467135x_{31} = -0.37235695467135
x32=−9.94375x_{32} = -9.94375
x33=−88.5928495612807x_{33} = -88.5928495612807
x34=−30.0935243403135x_{34} = -30.0935243403135
x35=−5.2x_{35} = -5.2
x36=−0.517236893456485x_{36} = -0.517236893456485
x37=−9.11627906976744x_{37} = -9.11627906976744
x38=−0.363870967741936x_{38} = -0.363870967741936
x39=−0.33347788378144x_{39} = -0.33347788378144
x40=−79.7948987616918x_{40} = -79.7948987616918
x41=−37.7840141037626x_{41} = -37.7840141037626
x42=−2x_{42} = -2
x43=−0.59447400034323x_{43} = -0.59447400034323
x44=−102.193141163736x_{44} = -102.193141163736
x45=−1.65552699228792x_{45} = -1.65552699228792
x46=−0.534592942484023x_{46} = -0.534592942484023
x47=−26.6508218713505x_{47} = -26.6508218713505
x48=−0.766562411145863x_{48} = -0.766562411145863
x49=−0.669565217391304x_{49} = -0.669565217391304
x50=−13.8928571428571x_{50} = -13.8928571428571
x51=−0.457819034989289x_{51} = -0.457819034989289
x52=−132x_{52} = -132
x53=−0.32665799739922x_{53} = -0.32665799739922
x54=−0.485711006541949x_{54} = -0.485711006541949
x55=−0.348010860583166x_{55} = -0.348010860583166
x56=−15.3176038962946x_{56} = -15.3176038962946
x57=−64x_{57} = -64
x58=−55.8037808764958x_{58} = -55.8037808764958
x59=−44.2178725094566x_{59} = -44.2178725094566
x60=−17.7825854700855x_{60} = -17.7825854700855
x61=−1.16129032258065x_{61} = -1.16129032258065
x62=−0.340589133261718x_{62} = -0.340589133261718
x63=−39.6587949693844x_{63} = -39.6587949693844
x64=−22.2326832175178x_{64} = -22.2326832175178
x65=−11.75x_{65} = -11.75
x66=−69.9484615622229x_{66} = -69.9484615622229
x67=−1.49524815205913x_{67} = -1.49524815205913
x68=−70.9685435151473x_{68} = -70.9685435151473
x69=−1.66931392299985x_{69} = -1.66931392299985
x70=−0.731227343345417x_{70} = -0.731227343345417
x71=−0.553159851301115x_{71} = -0.553159851301115
x72=−8x_{72} = -8
x73=−2.9618320610687x_{73} = -2.9618320610687
x74=−0.432965227708314x_{74} = -0.432965227708314
x75=−3.21569603451137x_{75} = -3.21569603451137
x76=−32x_{76} = -32
x77=−96.7506464290549x_{77} = -96.7506464290549
x78=−20.9790685239333x_{78} = -20.9790685239333
x79=−63.9585248953014x_{79} = -63.9585248953014
x80=−78.7056111291347x_{80} = -78.7056111291347
x81=−3.74566473988439x_{81} = -3.74566473988439
x82=−2.11475409836066x_{82} = -2.11475409836066
x83=−0.390577394664487x_{83} = -0.390577394664487
x84=−1.36395759717314x_{84} = -1.36395759717314
x85=−0.699029126213592x_{85} = -0.699029126213592
x86=−31.2102721785618x_{86} = -31.2102721785618
x87=−84.5242741388898x_{87} = -84.5242741388898
x88=−93.1209068262518x_{88} = -93.1209068262518
x89=−104.131879484838x_{89} = -104.131879484838
x90=−0.61754905590522x_{90} = -0.61754905590522
x91=−16x_{91} = -16
x92=−0.421526215608237x_{92} = -0.421526215608237
x93=−1.08132361189007x_{93} = -1.08132361189007
x94=−4x_{94} = -4
x95=−0.896819239720714x_{95} = -0.896819239720714
x96=−1.2543135783946x_{96} = -1.2543135783946
x97=−45.3505151872655x_{97} = -45.3505151872655
x98=−35.7477215029731x_{98} = -35.7477215029731
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 (|x/4 + 4/x| + x/4 + 4/x)/2.
(∣04+40∣+04)+402\frac{\left(\left|{\frac{0}{4} + \frac{4}{0}}\right| + \frac{0}{4}\right) + \frac{4}{0}}{2}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
(14−4x2)(x4+4x)2x216+2+16x2+18−2x2=0\frac{\left(\frac{1}{4} - \frac{4}{x^{2}}\right) \left(\frac{x}{4} + \frac{4}{x}\right)}{2 \sqrt{\frac{x^{2}}{16} + 2 + \frac{16}{x^{2}}}} + \frac{1}{8} - \frac{2}{x^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x4+∣x4+4x∣)+4x2)=0\lim_{x \to -\infty}\left(\frac{\left(\frac{x}{4} + \left|{\frac{x}{4} + \frac{4}{x}}\right|\right) + \frac{4}{x}}{2}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((x4+∣x4+4x∣)+4x2)=∞\lim_{x \to \infty}\left(\frac{\left(\frac{x}{4} + \left|{\frac{x}{4} + \frac{4}{x}}\right|\right) + \frac{4}{x}}{2}\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 (|x/4 + 4/x| + x/4 + 4/x)/2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x4+∣x4+4x∣)+4x2x)=0\lim_{x \to -\infty}\left(\frac{\left(\frac{x}{4} + \left|{\frac{x}{4} + \frac{4}{x}}\right|\right) + \frac{4}{x}}{2 x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((x4+∣x4+4x∣)+4x2x)=14\lim_{x \to \infty}\left(\frac{\left(\frac{x}{4} + \left|{\frac{x}{4} + \frac{4}{x}}\right|\right) + \frac{4}{x}}{2 x}\right) = \frac{1}{4}
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=x4y = \frac{x}{4}
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:
(x4+∣x4+4x∣)+4x2=−x8+∣x4+4x∣2−2x\frac{\left(\frac{x}{4} + \left|{\frac{x}{4} + \frac{4}{x}}\right|\right) + \frac{4}{x}}{2} = - \frac{x}{8} + \frac{\left|{\frac{x}{4} + \frac{4}{x}}\right|}{2} - \frac{2}{x}
- No
(x4+∣x4+4x∣)+4x2=x8−∣x4+4x∣2+2x\frac{\left(\frac{x}{4} + \left|{\frac{x}{4} + \frac{4}{x}}\right|\right) + \frac{4}{x}}{2} = \frac{x}{8} - \frac{\left|{\frac{x}{4} + \frac{4}{x}}\right|}{2} + \frac{2}{x}
- No
es decir, función
no es
par ni impar
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