Gráfico de la función y = xsin(pi/x)

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
$$\sin{\left(\frac{\pi}{x} \right)} - \frac{\pi \cos{\left(\frac{\pi}{x} \right)}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -42316.2298420376$$
$$x_{2} = -34688.1512445134$$
$$x_{3} = -30450.4076606265$$
$$x_{4} = 28039.0234837095$$
$$x_{5} = 40752.3171821719$$
$$x_{6} = -39773.5216497282$$
$$x_{7} = -33840.5969639233$$
$$x_{8} = 30581.6309983004$$
$$x_{9} = -20280.2966954216$$
$$x_{10} = 25496.4569159524$$
$$x_{11} = -37230.8277368372$$
$$x_{12} = 35666.9338247019$$
$$x_{13} = -32145.4963081903$$
$$x_{14} = 42447.4578596894$$
$$x_{15} = -27060.2754212728$$
$$x_{16} = -25365.2378549918$$
$$x_{17} = -21975.2418707685$$
$$x_{18} = -28755.3331458743$$
$$x_{19} = 39904.7490038284$$
$$x_{20} = -32993.0452507614$$
$$x_{21} = -26212.7538737752$$
$$x_{22} = -36383.2667900212$$
$$x_{23} = 20411.5078808788$$
$$x_{24} = 31429.1742172955$$
$$x_{25} = -38925.9552872708$$
$$x_{26} = 21258.9768989439$$
$$x_{27} = -21127.7639981051$$
$$x_{28} = 23801.4417704923$$
$$x_{29} = 41599.8868301447$$
$$x_{30} = -29602.8684848388$$
$$x_{31} = -23670.2247807833$$
$$x_{32} = 32276.7206419416$$
$$x_{33} = 34819.3768079445$$
$$x_{34} = 22953.9449120481$$
$$x_{35} = 39057.1823908182$$
$$x_{36} = 36514.493034162$$
$$x_{37} = 24648.9460175737$$
$$x_{38} = -22822.7291348599$$
$$x_{39} = 33971.8221480687$$
$$x_{40} = 26343.9738232498$$
$$x_{41} = -27907.8019934859$$
$$x_{42} = 33124.2700260645$$
$$x_{43} = -41468.6590201828$$
$$x_{44} = 27191.4961772367$$
$$x_{45} = -31297.9503613907$$
$$x_{46} = -35535.7079087597$$
$$x_{47} = 38209.6174473397$$
$$x_{48} = -38078.390611237$$
$$x_{49} = 29734.0912592147$$
$$x_{50} = 28886.5553065034$$
$$x_{51} = 37362.0542870497$$
$$x_{52} = -24517.7279386657$$
$$x_{53} = -40621.0895930176$$
$$x_{54} = 22106.4562927658$$
Signos de extremos en los puntos:

(-42316.22984203761, 42316.2298420376*sin(2.36315948687514e-5*pi))

(-34688.151244513436, 34688.1512445134*sin(2.8828287588782e-5*pi))

(-30450.407660626497, 30450.4076606265*sin(3.28402828344737e-5*pi))

(28039.02348370947, 28039.0234837095*sin(3.5664580137072e-5*pi))

(40752.317182171864, 40752.3171821719*sin(2.45384819599283e-5*pi))

(-39773.521649728194, 39773.5216497282*sin(2.51423549769281e-5*pi))

(-33840.5969639233, 33840.5969639233*sin(2.95503061327812e-5*pi))

(30581.63099830043, 30581.6309983004*sin(3.269936780205e-5*pi))

(-20280.296695421635, 20280.2966954216*sin(4.93089433068183e-5*pi))

(25496.456915952414, 25496.4569159524*sin(3.92211358345374e-5*pi))

(-37230.82773683721, 37230.8277368372*sin(2.68594619240918e-5*pi))

(35666.93382470188, 35666.9338247019*sin(2.80371731675861e-5*pi))

(-32145.496308190337, 32145.4963081903*sin(3.11085568694489e-5*pi))

(42447.45785968941, 42447.4578596894*sin(2.35585368458463e-5*pi))

(-27060.275421272778, 27060.2754212728*sin(3.69545388741267e-5*pi))

(-25365.237854991796, 25365.2378549918*sin(3.94240340152459e-5*pi))

(-21975.24187076853, 21975.2418707685*sin(4.5505756245177e-5*pi))

(-28755.33314587433, 28755.3331458743*sin(3.47761576931504e-5*pi))

(39904.74900382843, 39904.7490038284*sin(2.5059673972741e-5*pi))

(-32993.04525076136, 32993.0452507614*sin(3.03094180121771e-5*pi))

(-26212.75387377522, 26212.7538737752*sin(3.81493682356076e-5*pi))

(-36383.26679002124, 36383.2667900212*sin(2.74851624998739e-5*pi))

(20411.50788087885, 20411.5078808788*sin(4.89919708938693e-5*pi))

(31429.174217295455, 31429.1742172955*sin(3.18175715685747e-5*pi))

(-38925.955287270765, 38925.9552872708*sin(2.56897998422922e-5*pi))

(21258.97689894392, 21258.9768989439*sin(4.70389522860659e-5*pi))

(-21127.76399810513, 21127.7639981051*sin(4.73310853003511e-5*pi))

(23801.441770492336, 23801.4417704923*sin(4.20142615578752e-5*pi))

(41599.886830144715, 41599.8868301447*sin(2.40385269335725e-5*pi))

(-29602.868484838797, 29602.8684848388*sin(3.37805101729298e-5*pi))

(-23670.224780783272, 23670.2247807833*sin(4.22471695668836e-5*pi))

(32276.720641941552, 32276.7206419416*sin(3.09820818258892e-5*pi))

(34819.37680794453, 34819.3768079445*sin(2.87196409492268e-5*pi))

(22953.94491204806, 22953.9449120481*sin(4.35654962069339e-5*pi))

(39057.182390818234, 39057.1823908182*sin(2.56034854228267e-5*pi))

(36514.49303416201, 36514.493034162*sin(2.73863859773276e-5*pi))

(24648.94601757374, 24648.9460175737*sin(4.05696859933499e-5*pi))

(-22822.729134859903, 22822.7291348599*sin(4.38159693387667e-5*pi))

(33971.82214806872, 33971.8221480687*sin(2.94361602283629e-5*pi))

(26343.973823249846, 26343.9738232498*sin(3.79593453405823e-5*pi))

(-27907.801993485864, 27907.8019934859*sin(3.5832273721643e-5*pi))

(33124.27002606447, 33124.2700260645*sin(3.01893445263286e-5*pi))

(-41468.65902018279, 41468.6590201828*sin(2.4114596990303e-5*pi))

(27191.496177236688, 27191.4961772367*sin(3.6776203614612e-5*pi))

(-31297.950361390667, 31297.9503613907*sin(3.19509740559115e-5*pi))

(-35535.70790875974, 35535.7079087597*sin(2.8140708567494e-5*pi))

(38209.61744733968, 38209.6174473397*sin(2.61714214066183e-5*pi))

(-38078.390611236995, 38078.390611237*sin(2.62616141057416e-5*pi))

(29734.091259214732, 29734.0912592147*sin(3.36314297041143e-5*pi))

(28886.555306503415, 28886.5553065034*sin(3.4618180997679e-5*pi))

(37362.054287049716, 37362.0542870497*sin(2.6765123574766e-5*pi))

(-24517.727938665746, 24517.7279386657*sin(4.07868136273324e-5*pi))

(-40621.089593017634, 40621.0895930176*sin(2.46177542261666e-5*pi))

(22106.456292765764, 22106.4562927658*sin(4.52356536369534e-5*pi))

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:
$$x_{1} = 28039.0234837095$$
$$x_{2} = -32145.4963081903$$
$$x_{3} = 42447.4578596894$$
$$x_{4} = -26212.7538737752$$
$$x_{5} = 32276.7206419416$$
$$x_{6} = 34819.3768079445$$
$$x_{7} = 22953.9449120481$$
$$x_{8} = -35535.7079087597$$
Puntos máximos de la función:
$$x_{8} = -30450.4076606265$$
$$x_{8} = -21975.2418707685$$
$$x_{8} = 41599.8868301447$$
$$x_{8} = -23670.2247807833$$
$$x_{8} = 28886.5553065034$$
Decrece en los intervalos
$$\left[42447.4578596894, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -35535.7079087597\right]$$