Se da la desigualdad:
$$x \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} > \frac{1}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$x \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} = \frac{1}{2}$$
Resolvemos:
$$x_{1} = 18.9024837303424$$
$$x_{2} = 65.9582840466332$$
$$x_{3} = 21.9455654988197$$
$$x_{4} = 100.540911278755$$
$$x_{5} = -94.2583889420049$$
$$x_{6} = -81.6936501525055$$
$$x_{7} = 53.3883433351401$$
$$x_{8} = -69.12950448699$$
$$x_{9} = -97.3791029371005$$
$$x_{10} = 47.1026579842757$$
$$x_{11} = 69.12950448699$$
$$x_{12} = -2.77260470826599$$
$$x_{13} = 37.7256221319914$$
$$x_{14} = 15.6439973747731$$
$$x_{15} = -84.811210479617$$
$$x_{16} = -75.4114846544819$$
$$x_{17} = 87.9759612866821$$
$$x_{18} = -53.3883433351401$$
$$x_{19} = 59.6735017769259$$
$$x_{20} = 9.31724294141481$$
$$x_{21} = -62.8477652080076$$
$$x_{22} = -25.1724776117344$$
$$x_{23} = 50.2853702681169$$
$$x_{24} = -56.5663470412005$$
$$x_{25} = 1.11415714087193$$
$$x_{26} = -12.6455325787891$$
$$x_{27} = -40.8162019701941$$
$$x_{28} = -100.540911278755$$
$$x_{29} = -50.2853702681169$$
$$x_{30} = 81.6936501525055$$
$$x_{31} = -87.9759612866821$$
$$x_{32} = -103.662910767087$$
$$x_{33} = -18.9024837303424$$
$$x_{34} = 25.1724776117344$$
$$x_{35} = -44.0050237846841$$
$$x_{36} = 84.811210479617$$
$$x_{37} = -78.5270815352651$$
$$x_{38} = -72.2427883783755$$
$$x_{39} = 34.5285536017763$$
$$x_{40} = 56.5663470412005$$
$$x_{41} = 62.8477652080076$$
$$x_{42} = 12.6455325787891$$
$$x_{43} = 75.4114846544819$$
$$x_{44} = -15.6439973747731$$
$$x_{45} = -47.1026579842757$$
$$x_{46} = 31.4477306941599$$
$$x_{47} = -21.9455654988197$$
$$x_{48} = -59.6735017769259$$
$$x_{49} = -65.9582840466332$$
$$x_{50} = 97.3791029371005$$
$$x_{51} = 94.2583889420049$$
$$x_{52} = 28.238914350639$$
$$x_{53} = 44.0050237846841$$
$$x_{54} = -31.4477306941599$$
$$x_{55} = -9.31724294141481$$
$$x_{56} = -91.0952092079181$$
$$x_{57} = 78.5270815352651$$
$$x_{58} = 72.2427883783755$$
$$x_{59} = 91.0952092079181$$
$$x_{60} = -34.5285536017763$$
$$x_{61} = 2.77260470826599$$
$$x_{62} = -1.11415714087193$$
$$x_{63} = -37.7256221319914$$
$$x_{64} = -6.43911723841725$$
$$x_{65} = -28.238914350639$$
$$x_{66} = 40.8162019701941$$
$$x_{67} = 6.43911723841725$$
$$x_{1} = 18.9024837303424$$
$$x_{2} = 65.9582840466332$$
$$x_{3} = 21.9455654988197$$
$$x_{4} = 100.540911278755$$
$$x_{5} = -94.2583889420049$$
$$x_{6} = -81.6936501525055$$
$$x_{7} = 53.3883433351401$$
$$x_{8} = -69.12950448699$$
$$x_{9} = -97.3791029371005$$
$$x_{10} = 47.1026579842757$$
$$x_{11} = 69.12950448699$$
$$x_{12} = -2.77260470826599$$
$$x_{13} = 37.7256221319914$$
$$x_{14} = 15.6439973747731$$
$$x_{15} = -84.811210479617$$
$$x_{16} = -75.4114846544819$$
$$x_{17} = 87.9759612866821$$
$$x_{18} = -53.3883433351401$$
$$x_{19} = 59.6735017769259$$
$$x_{20} = 9.31724294141481$$
$$x_{21} = -62.8477652080076$$
$$x_{22} = -25.1724776117344$$
$$x_{23} = 50.2853702681169$$
$$x_{24} = -56.5663470412005$$
$$x_{25} = 1.11415714087193$$
$$x_{26} = -12.6455325787891$$
$$x_{27} = -40.8162019701941$$
$$x_{28} = -100.540911278755$$
$$x_{29} = -50.2853702681169$$
$$x_{30} = 81.6936501525055$$
$$x_{31} = -87.9759612866821$$
$$x_{32} = -103.662910767087$$
$$x_{33} = -18.9024837303424$$
$$x_{34} = 25.1724776117344$$
$$x_{35} = -44.0050237846841$$
$$x_{36} = 84.811210479617$$
$$x_{37} = -78.5270815352651$$
$$x_{38} = -72.2427883783755$$
$$x_{39} = 34.5285536017763$$
$$x_{40} = 56.5663470412005$$
$$x_{41} = 62.8477652080076$$
$$x_{42} = 12.6455325787891$$
$$x_{43} = 75.4114846544819$$
$$x_{44} = -15.6439973747731$$
$$x_{45} = -47.1026579842757$$
$$x_{46} = 31.4477306941599$$
$$x_{47} = -21.9455654988197$$
$$x_{48} = -59.6735017769259$$
$$x_{49} = -65.9582840466332$$
$$x_{50} = 97.3791029371005$$
$$x_{51} = 94.2583889420049$$
$$x_{52} = 28.238914350639$$
$$x_{53} = 44.0050237846841$$
$$x_{54} = -31.4477306941599$$
$$x_{55} = -9.31724294141481$$
$$x_{56} = -91.0952092079181$$
$$x_{57} = 78.5270815352651$$
$$x_{58} = 72.2427883783755$$
$$x_{59} = 91.0952092079181$$
$$x_{60} = -34.5285536017763$$
$$x_{61} = 2.77260470826599$$
$$x_{62} = -1.11415714087193$$
$$x_{63} = -37.7256221319914$$
$$x_{64} = -6.43911723841725$$
$$x_{65} = -28.238914350639$$
$$x_{66} = 40.8162019701941$$
$$x_{67} = 6.43911723841725$$
Las raíces dadas
$$x_{32} = -103.662910767087$$
$$x_{28} = -100.540911278755$$
$$x_{9} = -97.3791029371005$$
$$x_{5} = -94.2583889420049$$
$$x_{56} = -91.0952092079181$$
$$x_{31} = -87.9759612866821$$
$$x_{15} = -84.811210479617$$
$$x_{6} = -81.6936501525055$$
$$x_{37} = -78.5270815352651$$
$$x_{16} = -75.4114846544819$$
$$x_{38} = -72.2427883783755$$
$$x_{8} = -69.12950448699$$
$$x_{49} = -65.9582840466332$$
$$x_{21} = -62.8477652080076$$
$$x_{48} = -59.6735017769259$$
$$x_{24} = -56.5663470412005$$
$$x_{18} = -53.3883433351401$$
$$x_{29} = -50.2853702681169$$
$$x_{45} = -47.1026579842757$$
$$x_{35} = -44.0050237846841$$
$$x_{27} = -40.8162019701941$$
$$x_{63} = -37.7256221319914$$
$$x_{60} = -34.5285536017763$$
$$x_{54} = -31.4477306941599$$
$$x_{65} = -28.238914350639$$
$$x_{22} = -25.1724776117344$$
$$x_{47} = -21.9455654988197$$
$$x_{33} = -18.9024837303424$$
$$x_{44} = -15.6439973747731$$
$$x_{26} = -12.6455325787891$$
$$x_{55} = -9.31724294141481$$
$$x_{64} = -6.43911723841725$$
$$x_{12} = -2.77260470826599$$
$$x_{62} = -1.11415714087193$$
$$x_{25} = 1.11415714087193$$
$$x_{61} = 2.77260470826599$$
$$x_{67} = 6.43911723841725$$
$$x_{20} = 9.31724294141481$$
$$x_{42} = 12.6455325787891$$
$$x_{14} = 15.6439973747731$$
$$x_{1} = 18.9024837303424$$
$$x_{3} = 21.9455654988197$$
$$x_{34} = 25.1724776117344$$
$$x_{52} = 28.238914350639$$
$$x_{46} = 31.4477306941599$$
$$x_{39} = 34.5285536017763$$
$$x_{13} = 37.7256221319914$$
$$x_{66} = 40.8162019701941$$
$$x_{53} = 44.0050237846841$$
$$x_{10} = 47.1026579842757$$
$$x_{23} = 50.2853702681169$$
$$x_{7} = 53.3883433351401$$
$$x_{40} = 56.5663470412005$$
$$x_{19} = 59.6735017769259$$
$$x_{41} = 62.8477652080076$$
$$x_{2} = 65.9582840466332$$
$$x_{11} = 69.12950448699$$
$$x_{58} = 72.2427883783755$$
$$x_{43} = 75.4114846544819$$
$$x_{57} = 78.5270815352651$$
$$x_{30} = 81.6936501525055$$
$$x_{36} = 84.811210479617$$
$$x_{17} = 87.9759612866821$$
$$x_{59} = 91.0952092079181$$
$$x_{51} = 94.2583889420049$$
$$x_{50} = 97.3791029371005$$
$$x_{4} = 100.540911278755$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{32}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{32} - \frac{1}{10}$$
=
$$-103.662910767087 + - \frac{1}{10}$$
=
$$-103.762910767086$$
lo sustituimos en la expresión
$$x \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} > \frac{1}{2}$$
$$\left(-103.762910767086\right) \sin{\left(- \frac{103.762910767086}{2} \right)} \cos{\left(- \frac{103.762910767086}{2} \right)} > \frac{1}{2}$$
-4.68127994356077 > 1/2
Entonces
$$x < -103.662910767087$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -103.662910767087 \wedge x < -100.540911278755$$
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x32 x28 x9 x5 x56 x31 x15 x6 x37 x16 x38 x8 x49 x21 x48 x24 x18 x29 x45 x35 x27 x63 x60 x54 x65 x22 x47 x33 x44 x26 x55 x64 x12 x62 x25 x61 x67 x20 x42 x14 x1 x3 x34 x52 x46 x39 x13 x66 x53 x10 x23 x7 x40 x19 x41 x2 x11 x58 x43 x57 x30 x36 x17 x59 x51 x50 x4
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -103.662910767087 \wedge x < -100.540911278755$$
$$x > -97.3791029371005 \wedge x < -94.2583889420049$$
$$x > -91.0952092079181 \wedge x < -87.9759612866821$$
$$x > -84.811210479617 \wedge x < -81.6936501525055$$
$$x > -78.5270815352651 \wedge x < -75.4114846544819$$
$$x > -72.2427883783755 \wedge x < -69.12950448699$$
$$x > -65.9582840466332 \wedge x < -62.8477652080076$$
$$x > -59.6735017769259 \wedge x < -56.5663470412005$$
$$x > -53.3883433351401 \wedge x < -50.2853702681169$$
$$x > -47.1026579842757 \wedge x < -44.0050237846841$$
$$x > -40.8162019701941 \wedge x < -37.7256221319914$$
$$x > -34.5285536017763 \wedge x < -31.4477306941599$$
$$x > -28.238914350639 \wedge x < -25.1724776117344$$
$$x > -21.9455654988197 \wedge x < -18.9024837303424$$
$$x > -15.6439973747731 \wedge x < -12.6455325787891$$
$$x > -9.31724294141481 \wedge x < -6.43911723841725$$
$$x > -2.77260470826599 \wedge x < -1.11415714087193$$
$$x > 1.11415714087193 \wedge x < 2.77260470826599$$
$$x > 6.43911723841725 \wedge x < 9.31724294141481$$
$$x > 12.6455325787891 \wedge x < 15.6439973747731$$
$$x > 18.9024837303424 \wedge x < 21.9455654988197$$
$$x > 25.1724776117344 \wedge x < 28.238914350639$$
$$x > 31.4477306941599 \wedge x < 34.5285536017763$$
$$x > 37.7256221319914 \wedge x < 40.8162019701941$$
$$x > 44.0050237846841 \wedge x < 47.1026579842757$$
$$x > 50.2853702681169 \wedge x < 53.3883433351401$$
$$x > 56.5663470412005 \wedge x < 59.6735017769259$$
$$x > 62.8477652080076 \wedge x < 65.9582840466332$$
$$x > 69.12950448699 \wedge x < 72.2427883783755$$
$$x > 75.4114846544819 \wedge x < 78.5270815352651$$
$$x > 81.6936501525055 \wedge x < 84.811210479617$$
$$x > 87.9759612866821 \wedge x < 91.0952092079181$$
$$x > 94.2583889420049 \wedge x < 97.3791029371005$$
$$x > 100.540911278755$$