Sr Examen

Otras calculadoras

(x-3)(x-1)log[cos(pix)^2+cosx+2sin(x/2)^2,2^(1/2)/2]>=2 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
                   /                                  ______\     
                   |                                \/ 11/5 |     
                   |                        /   /x\\        |     
                   |                      2*|sin|-||        |     
                   |   2                    \   \2//        |     
(x - 3)*(x - 1)*log|cos (pi*x) + cos(x) + ------------------| >= 2
                   \                              2         /     
(x3)(x1)log((cos(x)+cos2(πx))+2sin115(x2)2)2\left(x - 3\right) \left(x - 1\right) \log{\left(\left(\cos{\left(x \right)} + \cos^{2}{\left(\pi x \right)}\right) + \frac{2 \sin^{\sqrt{\frac{11}{5}}}{\left(\frac{x}{2} \right)}}{2} \right)} \geq 2
((x - 3)*(x - 1))*log(cos(x) + cos(pi*x)^2 + (2*sin(x/2)^(sqrt(11/5)))/2) >= 2
Solución detallada
Se da la desigualdad:
(x3)(x1)log((cos(x)+cos2(πx))+2sin115(x2)2)2\left(x - 3\right) \left(x - 1\right) \log{\left(\left(\cos{\left(x \right)} + \cos^{2}{\left(\pi x \right)}\right) + \frac{2 \sin^{\sqrt{\frac{11}{5}}}{\left(\frac{x}{2} \right)}}{2} \right)} \geq 2
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
(x3)(x1)log((cos(x)+cos2(πx))+2sin115(x2)2)=2\left(x - 3\right) \left(x - 1\right) \log{\left(\left(\cos{\left(x \right)} + \cos^{2}{\left(\pi x \right)}\right) + \frac{2 \sin^{\sqrt{\frac{11}{5}}}{\left(\frac{x}{2} \right)}}{2} \right)} = 2
Resolvemos:
x1=100.5104354181340.0047674147312816ix_{1} = 100.510435418134 - 0.0047674147312816 i
x2=68.5297227402575x_{2} = 68.5297227402575
x3=93.65295473625190.0504357558232563ix_{3} = -93.6529547362519 - 0.0504357558232563 i
x4=52.2132777167091x_{4} = 52.2132777167091
x5=94.4430446648741+0.0239734809090849ix_{5} = 94.4430446648741 + 0.0239734809090849 i
x6=40.1269090033684x_{6} = 40.1269090033684
x7=33.8796199609159x_{7} = -33.8796199609159
x8=8.08522178928987+0.221804327440013ix_{8} = 8.08522178928987 + 0.221804327440013 i
x9=42.2523150363882x_{9} = 42.2523150363882
x10=87.4951623536677+0.0264918224471817ix_{10} = -87.4951623536677 + 0.0264918224471817 i
x11=61.6484065733173x_{11} = -61.6484065733173
x12=67.82710648852510.116532580524128ix_{12} = -67.8271064885251 - 0.116532580524128 i
x13=41.919402019937+0.253793657662776ix_{13} = -41.919402019937 + 0.253793657662776 i
x14=83.8209630426373x_{14} = -83.8209630426373
x15=47.8674926836918x_{15} = -47.8674926836918
x16=90.1676573854581x_{16} = 90.1676573854581
x17=16.9160112621997+0.236091194526466ix_{17} = -16.9160112621997 + 0.236091194526466 i
x18=7.69765946779217x_{18} = -7.69765946779217
x19=0.0259969878312562x_{19} = 0.0259969878312562
x20=58.11596529011640.168943046399089ix_{20} = 58.1159652901164 - 0.168943046399089 i
x21=28.0383674979943x_{21} = 28.0383674979943
x22=44.39122174487680.0386843113446253ix_{22} = 44.3912217448768 - 0.0386843113446253 i
x23=14.2557271704627x_{23} = 14.2557271704627
x24=57.6296158327081x_{24} = -57.6296158327081
x25=95.3640346092024x_{25} = -95.3640346092024
x26=95.6965840882506x_{26} = -95.6965840882506
x27=73.8767048669985+0.159634370423093ix_{27} = 73.8767048669985 + 0.159634370423093 i
x28=56.5037427905596+0.0142885202923343ix_{28} = 56.5037427905596 + 0.0142885202923343 i
x29=5.669327870230740.0497064137401943ix_{29} = -5.66932787023074 - 0.0497064137401943 i
x30=88.519795836584x_{30} = 88.519795836584
x31=26.3451765828346x_{31} = 26.3451765828346
x32=71.9446304817941x_{32} = -71.9446304817941
x33=92.1945249026872x_{33} = 92.1945249026872
x34=54.128574941131x_{34} = 54.128574941131
x35=81.54289719752770.0191245612540122ix_{35} = -81.5428971975277 - 0.0191245612540122 i
x36=15.9671278777495x_{36} = 15.9671278777495
x37=80.3190541652628x_{37} = 80.3190541652628
x38=73.7353154660035x_{38} = -73.7353154660035
x39=64.2998397574696x_{39} = 64.2998397574696
x40=35.90798723936270.212205057974673ix_{40} = 35.9079872393627 - 0.212205057974673 i
x41=52.0899566037580.217871757172483ix_{41} = -52.089956603758 - 0.217871757172483 i
x42=32.28024585873860.0691505085823875ix_{42} = 32.2802458587386 - 0.0691505085823875 i
x43=66.9212508083606+0.269489880500199ix_{43} = -66.9212508083606 + 0.269489880500199 i
x44=20.1647822182090.121405581267706ix_{44} = 20.164782218209 - 0.121405581267706 i
x45=92.7122815315705x_{45} = 92.7122815315705
x46=99.6012951400255x_{46} = -99.6012951400255
x47=6.385905190699540.00575895418760668ix_{47} = 6.38590519069954 - 0.00575895418760668 i
x48=54.89533003917680.186195356732807ix_{48} = -54.8953300391768 - 0.186195356732807 i
x49=31.49611063042420.0191707060545047ix_{49} = -31.4961106304242 - 0.0191707060545047 i
x50=14.11708445763110.166541967937758ix_{50} = -14.1170844576311 - 0.166541967937758 i
x51=23.6954544888424x_{51} = -23.6954544888424
x52=82.33321561590970.0540938198971391ix_{52} = 82.3332156159097 - 0.0540938198971391 i
x53=29.87950405943790.162851569333039ix_{53} = -29.8795040594379 - 0.162851569333039 i
x54=2.072542828419370.270802694872933ix_{54} = -2.07254282841937 - 0.270802694872933 i
x55=17.77429053084760.0886248084834091ix_{55} = -17.7742905308476 - 0.0886248084834091 i
x56=19.4587884353761x_{56} = -19.4587884353761
x57=59.9535607517084x_{57} = -59.9535607517084
x58=45.7553208896104x_{58} = -45.7553208896104
x59=111.823557484907+0.114305309917005ix_{59} = 111.823557484907 + 0.114305309917005 i
x60=50.49850713955530.0369607780358167ix_{60} = 50.4985071395553 - 0.0369607780358167 i
x61=76.3874647847242x_{61} = 76.3874647847242
x62=35.7814970140969x_{62} = -35.7814970140969
x63=11.72076716337650.0672315020436625ix_{63} = 11.7207671633765 - 0.0672315020436625 i
x64=49.4130262826674x_{64} = -49.4130262826674
x65=75.47345598046390.0125866265148061ix_{65} = -75.4734559804639 - 0.0125866265148061 i
x66=3.286586441332190.369342697073948ix_{66} = 3.28658644133219 - 0.369342697073948 i
x67=38.5957978455979x_{67} = 38.5957978455979
x68=5.30170266613166x_{68} = 5.30170266613166
x69=9.92049181202345x_{69} = -9.92049181202345
x70=85.91895225863730.251433509912749ix_{70} = 85.9189522586373 - 0.251433509912749 i
x71=25.41947723795310.0303405831403926ix_{71} = -25.4194772379531 - 0.0303405831403926 i
x72=55.71229761992280.0670079176742718ix_{72} = -55.7122976199228 - 0.0670079176742718 i
x73=37.5022710508658+0.0343942270405407ix_{73} = -37.5022710508658 + 0.0343942270405407 i
x74=18.49845927642670.0265263451148552ix_{74} = 18.4984592764267 - 0.0265263451148552 i
x75=43.60191028298490.0368709510682677ix_{75} = -43.6019102829849 - 0.0368709510682677 i
x76=11.6114441043886x_{76} = -11.6114441043886
x77=97.9075682228443x_{77} = -97.9075682228443
x78=70.21949368757210.0917905796213954ix_{78} = 70.2194936875721 - 0.0917905796213954 i
x79=80.7389286380715+0.075669195211865ix_{79} = -80.7389286380715 + 0.075669195211865 i
x80=40.90954550286560.321103912982183ix_{80} = -40.9095455028656 - 0.321103912982183 i
x81=85.8215792771042x_{81} = -85.8215792771042
x82=62.57268926952980.0287324695214719ix_{82} = 62.5726892695298 - 0.0287324695214719 i
x83=12.5437661962880.00146153110638568ix_{83} = 12.543766196288 - 0.00146153110638568 i
x84=78.081604889231x_{84} = 78.081604889231
Descartamos las soluciones complejas:
x1=68.5297227402575x_{1} = 68.5297227402575
x2=52.2132777167091x_{2} = 52.2132777167091
x3=40.1269090033684x_{3} = 40.1269090033684
x4=33.8796199609159x_{4} = -33.8796199609159
x5=42.2523150363882x_{5} = 42.2523150363882
x6=61.6484065733173x_{6} = -61.6484065733173
x7=83.8209630426373x_{7} = -83.8209630426373
x8=47.8674926836918x_{8} = -47.8674926836918
x9=90.1676573854581x_{9} = 90.1676573854581
x10=7.69765946779217x_{10} = -7.69765946779217
x11=0.0259969878312562x_{11} = 0.0259969878312562
x12=28.0383674979943x_{12} = 28.0383674979943
x13=14.2557271704627x_{13} = 14.2557271704627
x14=57.6296158327081x_{14} = -57.6296158327081
x15=95.3640346092024x_{15} = -95.3640346092024
x16=95.6965840882506x_{16} = -95.6965840882506
x17=88.519795836584x_{17} = 88.519795836584
x18=26.3451765828346x_{18} = 26.3451765828346
x19=71.9446304817941x_{19} = -71.9446304817941
x20=92.1945249026872x_{20} = 92.1945249026872
x21=54.128574941131x_{21} = 54.128574941131
x22=15.9671278777495x_{22} = 15.9671278777495
x23=80.3190541652628x_{23} = 80.3190541652628
x24=73.7353154660035x_{24} = -73.7353154660035
x25=64.2998397574696x_{25} = 64.2998397574696
x26=92.7122815315705x_{26} = 92.7122815315705
x27=99.6012951400255x_{27} = -99.6012951400255
x28=23.6954544888424x_{28} = -23.6954544888424
x29=19.4587884353761x_{29} = -19.4587884353761
x30=59.9535607517084x_{30} = -59.9535607517084
x31=45.7553208896104x_{31} = -45.7553208896104
x32=76.3874647847242x_{32} = 76.3874647847242
x33=35.7814970140969x_{33} = -35.7814970140969
x34=49.4130262826674x_{34} = -49.4130262826674
x35=38.5957978455979x_{35} = 38.5957978455979
x36=5.30170266613166x_{36} = 5.30170266613166
x37=9.92049181202345x_{37} = -9.92049181202345
x38=11.6114441043886x_{38} = -11.6114441043886
x39=97.9075682228443x_{39} = -97.9075682228443
x40=85.8215792771042x_{40} = -85.8215792771042
x41=78.081604889231x_{41} = 78.081604889231
Las raíces dadas
x27=99.6012951400255x_{27} = -99.6012951400255
x39=97.9075682228443x_{39} = -97.9075682228443
x16=95.6965840882506x_{16} = -95.6965840882506
x15=95.3640346092024x_{15} = -95.3640346092024
x40=85.8215792771042x_{40} = -85.8215792771042
x7=83.8209630426373x_{7} = -83.8209630426373
x24=73.7353154660035x_{24} = -73.7353154660035
x19=71.9446304817941x_{19} = -71.9446304817941
x6=61.6484065733173x_{6} = -61.6484065733173
x30=59.9535607517084x_{30} = -59.9535607517084
x14=57.6296158327081x_{14} = -57.6296158327081
x34=49.4130262826674x_{34} = -49.4130262826674
x8=47.8674926836918x_{8} = -47.8674926836918
x31=45.7553208896104x_{31} = -45.7553208896104
x33=35.7814970140969x_{33} = -35.7814970140969
x4=33.8796199609159x_{4} = -33.8796199609159
x28=23.6954544888424x_{28} = -23.6954544888424
x29=19.4587884353761x_{29} = -19.4587884353761
x38=11.6114441043886x_{38} = -11.6114441043886
x37=9.92049181202345x_{37} = -9.92049181202345
x10=7.69765946779217x_{10} = -7.69765946779217
x11=0.0259969878312562x_{11} = 0.0259969878312562
x36=5.30170266613166x_{36} = 5.30170266613166
x13=14.2557271704627x_{13} = 14.2557271704627
x22=15.9671278777495x_{22} = 15.9671278777495
x18=26.3451765828346x_{18} = 26.3451765828346
x12=28.0383674979943x_{12} = 28.0383674979943
x35=38.5957978455979x_{35} = 38.5957978455979
x3=40.1269090033684x_{3} = 40.1269090033684
x5=42.2523150363882x_{5} = 42.2523150363882
x2=52.2132777167091x_{2} = 52.2132777167091
x21=54.128574941131x_{21} = 54.128574941131
x25=64.2998397574696x_{25} = 64.2998397574696
x1=68.5297227402575x_{1} = 68.5297227402575
x32=76.3874647847242x_{32} = 76.3874647847242
x41=78.081604889231x_{41} = 78.081604889231
x23=80.3190541652628x_{23} = 80.3190541652628
x17=88.519795836584x_{17} = 88.519795836584
x9=90.1676573854581x_{9} = 90.1676573854581
x20=92.1945249026872x_{20} = 92.1945249026872
x26=92.7122815315705x_{26} = 92.7122815315705
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0x27x_{0} \leq x_{27}
Consideremos, por ejemplo, el punto
x0=x27110x_{0} = x_{27} - \frac{1}{10}
=
99.6012951400255+110-99.6012951400255 + - \frac{1}{10}
=
99.7012951400255-99.7012951400255
lo sustituimos en la expresión
(x3)(x1)log((cos(x)+cos2(πx))+2sin115(x2)2)2\left(x - 3\right) \left(x - 1\right) \log{\left(\left(\cos{\left(x \right)} + \cos^{2}{\left(\pi x \right)}\right) + \frac{2 \sin^{\sqrt{\frac{11}{5}}}{\left(\frac{x}{2} \right)}}{2} \right)} \geq 2
(99.70129514002553)(99.70129514002551)log(2sin115(99.70129514002552)2+(cos2((99.7012951400255)π)+cos(99.7012951400255)))2\left(-99.7012951400255 - 3\right) \left(-99.7012951400255 - 1\right) \log{\left(\frac{2 \sin^{\sqrt{\frac{11}{5}}}{\left(- \frac{99.7012951400255}{2} \right)}}{2} + \left(\cos^{2}{\left(\left(-99.7012951400255\right) \pi \right)} + \cos{\left(-99.7012951400255 \right)}\right) \right)} \geq 2
                    /                                       ____                            \     
                    |                                     \/ 55                             |     
                    |                                     ------                            | >= 2
                    |                                       5         2                     |     
10342.1534331586*log\0.675119406768933 + 0.403038827677351       + cos (1.70129514002545*pi)/     

significa que una de las soluciones de nuestra ecuación será con:
x99.6012951400255x \leq -99.6012951400255
 _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____          
      \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \    
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
       x27      x39      x16      x15      x40      x7      x24      x19      x6      x30      x14      x34      x8      x31      x33      x4      x28      x29      x38      x37      x10      x11      x36      x13      x22      x18      x12      x35      x3      x5      x2      x21      x25      x1      x32      x41      x23      x17      x9      x20      x26

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x99.6012951400255x \leq -99.6012951400255
x97.9075682228443x95.6965840882506x \geq -97.9075682228443 \wedge x \leq -95.6965840882506
x95.3640346092024x85.8215792771042x \geq -95.3640346092024 \wedge x \leq -85.8215792771042
x83.8209630426373x73.7353154660035x \geq -83.8209630426373 \wedge x \leq -73.7353154660035
x71.9446304817941x61.6484065733173x \geq -71.9446304817941 \wedge x \leq -61.6484065733173
x59.9535607517084x57.6296158327081x \geq -59.9535607517084 \wedge x \leq -57.6296158327081
x49.4130262826674x47.8674926836918x \geq -49.4130262826674 \wedge x \leq -47.8674926836918
x45.7553208896104x35.7814970140969x \geq -45.7553208896104 \wedge x \leq -35.7814970140969
x33.8796199609159x23.6954544888424x \geq -33.8796199609159 \wedge x \leq -23.6954544888424
x19.4587884353761x11.6114441043886x \geq -19.4587884353761 \wedge x \leq -11.6114441043886
x9.92049181202345x7.69765946779217x \geq -9.92049181202345 \wedge x \leq -7.69765946779217
x0.0259969878312562x5.30170266613166x \geq 0.0259969878312562 \wedge x \leq 5.30170266613166
x14.2557271704627x15.9671278777495x \geq 14.2557271704627 \wedge x \leq 15.9671278777495
x26.3451765828346x28.0383674979943x \geq 26.3451765828346 \wedge x \leq 28.0383674979943
x38.5957978455979x40.1269090033684x \geq 38.5957978455979 \wedge x \leq 40.1269090033684
x42.2523150363882x52.2132777167091x \geq 42.2523150363882 \wedge x \leq 52.2132777167091
x54.128574941131x64.2998397574696x \geq 54.128574941131 \wedge x \leq 64.2998397574696
x68.5297227402575x76.3874647847242x \geq 68.5297227402575 \wedge x \leq 76.3874647847242
x78.081604889231x80.3190541652628x \geq 78.081604889231 \wedge x \leq 80.3190541652628
x88.519795836584x90.1676573854581x \geq 88.519795836584 \wedge x \leq 90.1676573854581
x92.1945249026872x92.7122815315705x \geq 92.1945249026872 \wedge x \leq 92.7122815315705
Solución de la desigualdad en el gráfico
05-30-25-20-15-10-5101520253035-50005000