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sin(x)^(2)*cos(x)^(2)
Trazado el gráfico de la función/ sin(x)^(2)*cos(x)^(2)

Gráfico de la función y = sin(x)^(2)*cos(x)^(2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2       2   
f(x) = sin (x)*cos (x)
f(x)=sin⁡2(x)cos⁡2(x)f{\left(x \right)} = \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} 
f = sin(x)^2*cos(x)^2
Gráfico de la función
02468-8-6-4-2-10100.000.50
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:
sin⁡2(x)cos⁡2(x)=0\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Solución numérica
x1=−6.28318518328035x_{1} = -6.28318518328035
x2=14.1371670778185x_{2} = 14.1371670778185
x3=12.5663704969137x_{3} = 12.5663704969137
x4=−81.6814090370675x_{4} = -81.6814090370675
x5=−89.5353907315491x_{5} = -89.5353907315491
x6=−59.6902604569585x_{6} = -59.6902604569585
x7=37.6991119665793x_{7} = 37.6991119665793
x8=−97.3893725907902x_{8} = -97.3893725907902
x9=−75.3982237985682x_{9} = -75.3982237985682
x10=−37.6991118766796x_{10} = -37.6991118766796
x11=78.5398162225044x_{11} = 78.5398162225044
x12=−1.57079626356835x_{12} = -1.57079626356835
x13=6.28318528443138x_{13} = 6.28318528443138
x14=26.7035375390573x_{14} = 26.7035375390573
x15=−7.85398150696156x_{15} = -7.85398150696156
x16=−83.2522051669813x_{16} = -83.2522051669813
x17=−97.3893723711949x_{17} = -97.3893723711949
x18=−86.3937978789102x_{18} = -86.3937978789102
x19=−100.530965206253x_{19} = -100.530965206253
x20=15.7079633917898x_{20} = 15.7079633917898
x21=50.2654824463816x_{21} = 50.2654824463816
x22=−29.8451301000724x_{22} = -29.8451301000724
x23=−21.9911485864927x_{23} = -21.9911485864927
x24=−95.818575868455x_{24} = -95.818575868455
x25=−72.2566309100272x_{25} = -72.2566309100272
x26=−53.4070752253874x_{26} = -53.4070752253874
x27=23.5619449483644x_{27} = 23.5619449483644
x28=45.5530935075531x_{28} = 45.5530935075531
x29=−64.402649310466x_{29} = -64.402649310466
x30=−28.2743337586152x_{30} = -28.2743337586152
x31=80.1106131511482x_{31} = 80.1106131511482
x32=−42.4115007432387x_{32} = -42.4115007432387
x33=64.4026493150839x_{33} = 64.4026493150839
x34=−67.5442421539445x_{34} = -67.5442421539445
x35=34.5575190717885x_{35} = 34.5575190717885
x36=−45.5530935761698x_{36} = -45.5530935761698
x37=0x_{37} = 0
x38=65.9734457525462x_{38} = 65.9734457525462
x39=−87.9645943594276x_{39} = -87.9645943594276
x40=59.690260541069x_{40} = 59.690260541069
x41=28.2743338652921x_{41} = 28.2743338652921
x42=86.3937978937855x_{42} = 86.3937978937855
x43=29.8451303084991x_{43} = 29.8451303084991
x44=95.8185760424586x_{44} = 95.8185760424586
x45=92.6769832182628x_{45} = 92.6769832182628
x46=20.4203521581227x_{46} = 20.4203521581227
x47=48.6946860958663x_{47} = 48.6946860958663
x48=81.6814091152362x_{48} = 81.6814091152362
x49=−43.9822971747455x_{49} = -43.9822971747455
x50=−17.2787595621355x_{50} = -17.2787595621355
x51=−51.8362786915081x_{51} = -51.8362786915081
x52=−1.57079642013166x_{52} = -1.57079642013166
x53=4.71238898608896x_{53} = 4.71238898608896
x54=70.6858338406532x_{54} = 70.6858338406532
x55=−50.2654823342013x_{55} = -50.2654823342013
x56=−58.1194640062544x_{56} = -58.1194640062544
x57=67.5442420634706x_{57} = 67.5442420634706
x58=−15.7079632962205x_{58} = -15.7079632962205
x59=−73.8274272808521x_{59} = -73.8274272808521
x60=73.8274274646672x_{60} = 73.8274274646672
x61=−39.2699081045218x_{61} = -39.2699081045218
x62=21.9911485851564x_{62} = 21.9911485851564
x63=95.8185756842062x_{63} = 95.8185756842062
x64=56.5486676469942x_{64} = 56.5486676469942
x65=42.4115007365289x_{65} = 42.4115007365289
x66=−94.247779486083x_{66} = -94.247779486083
x67=89.5353906153414x_{67} = 89.5353906153414
x68=−14.1371668484631x_{68} = -14.1371668484631
x69=−20.4203521774723x_{69} = -20.4203521774723
x70=43.9822971692691x_{70} = 43.9822971692691
x71=36.1283154718409x_{71} = 36.1283154718409
x72=72.2566310277248x_{72} = 72.2566310277248
x73=−80.1106125854791x_{73} = -80.1106125854791
x74=−58.1194645366003x_{74} = -58.1194645366003
x75=−23.5619449982306x_{75} = -23.5619449982306
x76=7.85398173011892x_{76} = 7.85398173011892
x77=51.8362788866811x_{77} = 51.8362788866811
x78=100.530964798296x_{78} = 100.530964798296
x79=87.9645943351391x_{79} = 87.9645943351391
x80=−36.128315427252x_{80} = -36.128315427252
x81=94.247779609353x_{81} = 94.247779609353
x82=−65.9734457653935x_{82} = -65.9734457653935
x83=1.57079638652515x_{83} = 1.57079638652515
x84=72.256630710694x_{84} = 72.256630710694
x85=−31.4159266517141x_{85} = -31.4159266517141
x86=−61.2610566398387x_{86} = -61.2610566398387
x87=117.809724442492x_{87} = 117.809724442492
x88=−9.42477807759933x_{88} = -9.42477807759933
x89=70.6858346557926x_{89} = 70.6858346557926
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
−2sin⁡3(x)cos⁡(x)+2sin⁡(x)cos⁡3(x)=0- 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 2 \sin{\left(x \right)} \cos^{3}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−3π4x_{2} = - \frac{3 \pi}{4}
x3=−π2x_{3} = - \frac{\pi}{2}
x4=−π4x_{4} = - \frac{\pi}{4}
x5=π4x_{5} = \frac{\pi}{4}
x6=π2x_{6} = \frac{\pi}{2}
x7=3π4x_{7} = \frac{3 \pi}{4}
Signos de extremos en los puntos:
(0, 0)

 -3*pi      
(-----, 1/4)
   4        

 -pi     
(----, 0)
  2      

 -pi       
(----, 1/4)
  4        

 pi      
(--, 1/4)
 4       

 pi    
(--, 0)
 2     

 3*pi      
(----, 1/4)
  4


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=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Puntos máximos de la función:
x3=−3π4x_{3} = - \frac{3 \pi}{4}
x3=−π4x_{3} = - \frac{\pi}{4}
x3=π4x_{3} = \frac{\pi}{4}
x3=3π4x_{3} = \frac{3 \pi}{4}
Decrece en los intervalos
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Crece en los intervalos
(−∞,−π2]\left(-\infty, - \frac{\pi}{2}\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((sin⁡2(x)−cos⁡2(x))sin⁡2(x)−(sin⁡2(x)−cos⁡2(x))cos⁡2(x)−4sin⁡2(x)cos⁡2(x))=02 \left(\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} - 4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−2atan⁡(−1+22−2+2)x_{1} = - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x2=2atan⁡(−1+22−2+2)x_{2} = 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x3=−2atan⁡(1+2+22+2)x_{3} = - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x4=2atan⁡(1+2+22+2)x_{4} = 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x5=−2atan⁡(−2+1+22−2)x_{5} = - 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x6=2atan⁡(−2+1+22−2)x_{6} = 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x7=−2atan⁡(−22+2+1+2)x_{7} = - 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}
x8=2atan⁡(−22+2+1+2)x_{8} = 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[2atan⁡(1+2+22+2),∞)\left[2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}, \infty\right)
Convexa en los intervalos
(−∞,−2atan⁡(−1+22−2+2)]\left(-\infty, - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}\right]
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)^2*cos(x)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡2(x)cos⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡2(x)cos⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
sin⁡2(x)cos⁡2(x)=sin⁡2(x)cos⁡2(x)\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}
- Sí
sin⁡2(x)cos⁡2(x)=−sin⁡2(x)cos⁡2(x)\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = sin(x)^(2)*cos(x)^(2)
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