Mister Exam
Graphing y = cos(x)*sin(x^2)
The solution
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/ 2\ f(x) = cos(x)*sin\x /
$$f{\left(x \right)} = \sin{\left(x^{2} \right)} \cos{\left(x \right)}$$
f = sin(x^2)*cos(x)
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\sin{\left(x^{2} \right)} \cos{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = -45.8103033454045$$
$$x_{2} = -98.9562746936211$$
$$x_{3} = -86.1602195936888$$
$$x_{4} = -73.9349120324508$$
$$x_{5} = -27.6298211145552$$
$$x_{6} = -69.8490758065198$$
$$x_{7} = 3.96332729760601$$
$$x_{8} = -91.9628589348098$$
$$x_{9} = 64.2499240996983$$
$$x_{10} = -14.0684162995043$$
$$x_{11} = 59.8449298194288$$
$$x_{12} = -92.6774746899941$$
$$x_{13} = -51.8575289757704$$
$$x_{14} = -48.9915991506973$$
$$x_{15} = 73.9986215763984$$
$$x_{16} = 94.1071181711013$$
$$x_{17} = 22.137941502317$$
$$x_{18} = -23.5142575266965$$
$$x_{19} = 28.2482660354898$$
$$x_{20} = 73.1875363321556$$
$$x_{21} = 10.7814158709709$$
$$x_{22} = -29.8172889607005$$
$$x_{23} = -89.7498945058111$$
$$x_{24} = 16.244807875181$$
$$x_{25} = 8.1224039375905$$
$$x_{26} = -58.0055663880172$$
$$x_{27} = -31.2072703486357$$
$$x_{28} = 80.0945204032733$$
$$x_{29} = -55.1173302693566$$
$$x_{30} = 61.2715531143107$$
$$x_{31} = 76.9690200129499$$
$$x_{32} = -23.5619449019235$$
$$x_{33} = 10.6347231054331$$
$$x_{34} = 95.8109270062104$$
$$x_{35} = -84.447707628146$$
$$x_{36} = -89.5396250380866$$
$$x_{37} = 14.1371669411541$$
$$x_{38} = -36.3677124268396$$
$$x_{39} = 52.5794782701295$$
$$x_{40} = 77.5840932426103$$
$$x_{41} = -11.7571287633483$$
$$x_{42} = -5.87856438167413$$
$$x_{43} = -15.7539144225679$$
$$x_{44} = 101.526181334807$$
$$x_{45} = 58.1408092071534$$
$$x_{46} = 86.2331131935235$$
$$x_{47} = 98.1273737713015$$
$$x_{48} = 42.2052488985128$$
$$x_{49} = 0$$
$$x_{50} = 82.2235209081029$$
$$x_{51} = 89.5571663498502$$
$$x_{52} = -391.210720344267$$
$$x_{53} = -22.0668724858422$$
$$x_{54} = -65.6765309176633$$
$$x_{55} = -9.86860538583257$$
$$x_{56} = 36.1511070908396$$
$$x_{57} = -61.261056745001$$
$$x_{58} = 66.2006174575265$$
$$x_{59} = -64.2743676707175$$
$$x_{60} = -67.678963415037$$
$$x_{61} = -23.8459277508708$$
$$x_{62} = 52.2498231190263$$
$$x_{63} = 39.2349385147855$$
$$x_{64} = 18.2485292908913$$
$$x_{65} = -55.8815094433593$$
$$x_{66} = -70.6540213049065$$
$$x_{67} = 20.209083229248$$
$$x_{68} = 23.7799637856361$$
$$x_{69} = -41.793845424846$$
$$x_{70} = 46.3217851815086$$
$$x_{71} = 61.2971843863435$$
$$x_{72} = -1.77245385090552$$
$$x_{73} = 83.6815990815881$$
$$x_{74} = 70.6984717788253$$
$$x_{75} = 6.13996024767893$$
$$x_{76} = 66.6263492885688$$
$$x_{77} = 78.9289441970977$$
$$x_{78} = 46.4234053076958$$
$$x_{79} = -17.9009064202391$$
$$x_{80} = -7.72594721818665$$
$$x_{81} = 30.079539295572$$
$$x_{82} = -95.8185759344887$$
$$x_{83} = -43.8480866628973$$
$$x_{84} = -34.139872209512$$
$$x_{85} = -59.9236214113694$$
$$x_{86} = -19.8166364880301$$
$$x_{87} = 1.77245385090552$$
$$x_{88} = 39.1547853391105$$
$$x_{89} = -25.8073176165412$$
$$x_{90} = 32.9867228626928$$
$$x_{91} = -39.751994978311$$
$$x_{92} = 45.535163899664$$
$$x_{93} = -77.5233303355659$$
$$x_{94} = -33.862683274665$$
$$x_{95} = -76.8926353694292$$
$$x_{96} = -3.96332729760601$$
$$x_{97} = -98.9721470611453$$
$$x_{98} = 92.6769832808989$$
$$x_{99} = 54.9778714378214$$
$$x_{100} = -83.8503688646267$$
so we need to solve the equation:
$$\sin{\left(x^{2} \right)} \cos{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = -45.8103033454045$$
$$x_{2} = -98.9562746936211$$
$$x_{3} = -86.1602195936888$$
$$x_{4} = -73.9349120324508$$
$$x_{5} = -27.6298211145552$$
$$x_{6} = -69.8490758065198$$
$$x_{7} = 3.96332729760601$$
$$x_{8} = -91.9628589348098$$
$$x_{9} = 64.2499240996983$$
$$x_{10} = -14.0684162995043$$
$$x_{11} = 59.8449298194288$$
$$x_{12} = -92.6774746899941$$
$$x_{13} = -51.8575289757704$$
$$x_{14} = -48.9915991506973$$
$$x_{15} = 73.9986215763984$$
$$x_{16} = 94.1071181711013$$
$$x_{17} = 22.137941502317$$
$$x_{18} = -23.5142575266965$$
$$x_{19} = 28.2482660354898$$
$$x_{20} = 73.1875363321556$$
$$x_{21} = 10.7814158709709$$
$$x_{22} = -29.8172889607005$$
$$x_{23} = -89.7498945058111$$
$$x_{24} = 16.244807875181$$
$$x_{25} = 8.1224039375905$$
$$x_{26} = -58.0055663880172$$
$$x_{27} = -31.2072703486357$$
$$x_{28} = 80.0945204032733$$
$$x_{29} = -55.1173302693566$$
$$x_{30} = 61.2715531143107$$
$$x_{31} = 76.9690200129499$$
$$x_{32} = -23.5619449019235$$
$$x_{33} = 10.6347231054331$$
$$x_{34} = 95.8109270062104$$
$$x_{35} = -84.447707628146$$
$$x_{36} = -89.5396250380866$$
$$x_{37} = 14.1371669411541$$
$$x_{38} = -36.3677124268396$$
$$x_{39} = 52.5794782701295$$
$$x_{40} = 77.5840932426103$$
$$x_{41} = -11.7571287633483$$
$$x_{42} = -5.87856438167413$$
$$x_{43} = -15.7539144225679$$
$$x_{44} = 101.526181334807$$
$$x_{45} = 58.1408092071534$$
$$x_{46} = 86.2331131935235$$
$$x_{47} = 98.1273737713015$$
$$x_{48} = 42.2052488985128$$
$$x_{49} = 0$$
$$x_{50} = 82.2235209081029$$
$$x_{51} = 89.5571663498502$$
$$x_{52} = -391.210720344267$$
$$x_{53} = -22.0668724858422$$
$$x_{54} = -65.6765309176633$$
$$x_{55} = -9.86860538583257$$
$$x_{56} = 36.1511070908396$$
$$x_{57} = -61.261056745001$$
$$x_{58} = 66.2006174575265$$
$$x_{59} = -64.2743676707175$$
$$x_{60} = -67.678963415037$$
$$x_{61} = -23.8459277508708$$
$$x_{62} = 52.2498231190263$$
$$x_{63} = 39.2349385147855$$
$$x_{64} = 18.2485292908913$$
$$x_{65} = -55.8815094433593$$
$$x_{66} = -70.6540213049065$$
$$x_{67} = 20.209083229248$$
$$x_{68} = 23.7799637856361$$
$$x_{69} = -41.793845424846$$
$$x_{70} = 46.3217851815086$$
$$x_{71} = 61.2971843863435$$
$$x_{72} = -1.77245385090552$$
$$x_{73} = 83.6815990815881$$
$$x_{74} = 70.6984717788253$$
$$x_{75} = 6.13996024767893$$
$$x_{76} = 66.6263492885688$$
$$x_{77} = 78.9289441970977$$
$$x_{78} = 46.4234053076958$$
$$x_{79} = -17.9009064202391$$
$$x_{80} = -7.72594721818665$$
$$x_{81} = 30.079539295572$$
$$x_{82} = -95.8185759344887$$
$$x_{83} = -43.8480866628973$$
$$x_{84} = -34.139872209512$$
$$x_{85} = -59.9236214113694$$
$$x_{86} = -19.8166364880301$$
$$x_{87} = 1.77245385090552$$
$$x_{88} = 39.1547853391105$$
$$x_{89} = -25.8073176165412$$
$$x_{90} = 32.9867228626928$$
$$x_{91} = -39.751994978311$$
$$x_{92} = 45.535163899664$$
$$x_{93} = -77.5233303355659$$
$$x_{94} = -33.862683274665$$
$$x_{95} = -76.8926353694292$$
$$x_{96} = -3.96332729760601$$
$$x_{97} = -98.9721470611453$$
$$x_{98} = 92.6769832808989$$
$$x_{99} = 54.9778714378214$$
$$x_{100} = -83.8503688646267$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$2 x \cos{\left(x \right)} \cos{\left(x^{2} \right)} - \sin{\left(x \right)} \sin{\left(x^{2} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -21.7443208246461$$
$$x_{2} = -57.6114316103265$$
$$x_{3} = 66.0699800161888$$
$$x_{4} = 56.175913886337$$
$$x_{5} = -54.2410486649869$$
$$x_{6} = 20.4309423892029$$
$$x_{7} = -36.0818579563209$$
$$x_{8} = 29.829741482418$$
$$x_{9} = 82.7850976847909$$
$$x_{10} = 78.1389021841115$$
$$x_{11} = -10.957767460846$$
$$x_{12} = 62.2002992999418$$
$$x_{13} = -36.1766711135295$$
$$x_{14} = 98.967318895804$$
$$x_{15} = -14.0032317322368$$
$$x_{16} = -42.0743813758713$$
$$x_{17} = 46.032869960288$$
$$x_{18} = 64.3994668189113$$
$$x_{19} = 26.7041238409344$$
$$x_{20} = -29.3120570723519$$
$$x_{21} = 22.3144555829732$$
$$x_{22} = 84.2894731854299$$
$$x_{23} = -42.4203679035566$$
$$x_{24} = -71.8554832810555$$
$$x_{25} = 42.1860239023499$$
$$x_{26} = -16.0009491904737$$
$$x_{27} = -53.7465455610153$$
$$x_{28} = -67.6216177692932$$
$$x_{29} = -62.0232911735757$$
$$x_{30} = 34.2088961514732$$
$$x_{31} = 51.8487347950227$$
$$x_{32} = -79.8291866176748$$
$$x_{33} = -23.749242427506$$
$$x_{34} = 7.7808229418531$$
$$x_{35} = -62.8784063418455$$
$$x_{36} = 4.70091565676332$$
$$x_{37} = 26.259086539065$$
$$x_{38} = 10.2562149360732$$
$$x_{39} = 36.1408978109486$$
$$x_{40} = -3.74766067133477$$
$$x_{41} = 82.461938457206$$
$$x_{42} = 60.2503535867024$$
$$x_{43} = -26.4373808099425$$
$$x_{44} = 61.1809301963892$$
$$x_{45} = 18.0328686012482$$
$$x_{46} = -65.8556721673776$$
$$x_{47} = -33.7001911481362$$
$$x_{48} = -51.7503243924615$$
$$x_{49} = 94.8966350594679$$
$$x_{50} = 86.7145856729587$$
$$x_{51} = 96.0811603317475$$
$$x_{52} = 23.6214725471253$$
$$x_{53} = 0$$
$$x_{54} = 90.3169469354969$$
$$x_{55} = -92.8385290764838$$
$$x_{56} = -71.0200484194458$$
$$x_{57} = -47.9381265468237$$
$$x_{58} = -45.5148417861163$$
$$x_{59} = -44.0803123955744$$
$$x_{60} = -85.8039257747406$$
$$x_{61} = -1.67484432484861$$
$$x_{62} = 54.9762522924857$$
$$x_{63} = -67.5251887322074$$
$$x_{64} = 51.8088129558038$$
$$x_{65} = -95.8526961740184$$
$$x_{66} = -98.8528049848859$$
$$x_{67} = 76.4315416810569$$
$$x_{68} = 39.3763165001392$$
$$x_{69} = 73.8280180700468$$
$$x_{70} = 4.13465262627828$$
$$x_{71} = 14.2458403046005$$
$$x_{72} = 15.9025787613332$$
$$x_{73} = -83.972087632902$$
$$x_{74} = -8.22787376837475$$
$$x_{75} = -97.7343881376139$$
$$x_{76} = -93.881522316429$$
$$x_{77} = 47.7411566321408$$
$$x_{78} = -77.8367964156153$$
$$x_{79} = -75.8125376133442$$
$$x_{80} = -82.1183644934327$$
$$x_{81} = 6.26667579623329$$
$$x_{82} = 2.23412500791198$$
$$x_{83} = -19.9339927192786$$
$$x_{84} = -31.8794194517317$$
$$x_{85} = -90.2995543164443$$
$$x_{86} = -7.7808229418531$$
$$x_{87} = 99.0596546447822$$
$$x_{88} = -59.6740765195264$$
$$x_{89} = 67.5251887322074$$
$$x_{90} = 92.634412032881$$
$$x_{91} = -55.8393982203108$$
$$x_{92} = -81.8885110144078$$
$$x_{93} = 86.4081996571392$$
$$x_{94} = -73.7952790938783$$
$$x_{95} = 37.536735170322$$
$$x_{96} = -17.2772507335565$$
$$x_{97} = -5.7478922734498$$
$$x_{98} = -17.7702721697402$$
$$x_{99} = 26.7424722578364$$
The values of the extrema at the points:
Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = -21.7443208246461$$
$$x_{2} = -54.2410486649869$$
$$x_{3} = 20.4309423892029$$
$$x_{4} = -36.0818579563209$$
$$x_{5} = 82.7850976847909$$
$$x_{6} = -10.957767460846$$
$$x_{7} = 62.2002992999418$$
$$x_{8} = 98.967318895804$$
$$x_{9} = 46.032869960288$$
$$x_{10} = 26.7041238409344$$
$$x_{11} = 22.3144555829732$$
$$x_{12} = 42.1860239023499$$
$$x_{13} = -67.6216177692932$$
$$x_{14} = 34.2088961514732$$
$$x_{15} = -79.8291866176748$$
$$x_{16} = -23.749242427506$$
$$x_{17} = 7.7808229418531$$
$$x_{18} = 26.259086539065$$
$$x_{19} = 36.1408978109486$$
$$x_{20} = -3.74766067133477$$
$$x_{21} = 18.0328686012482$$
$$x_{22} = -65.8556721673776$$
$$x_{23} = 86.7145856729587$$
$$x_{24} = 96.0811603317475$$
$$x_{25} = 23.6214725471253$$
$$x_{26} = 0$$
$$x_{27} = 90.3169469354969$$
$$x_{28} = -92.8385290764838$$
$$x_{29} = -45.5148417861163$$
$$x_{30} = -1.67484432484861$$
$$x_{31} = 54.9762522924857$$
$$x_{32} = -95.8526961740184$$
$$x_{33} = -98.8528049848859$$
$$x_{34} = 76.4315416810569$$
$$x_{35} = 14.2458403046005$$
$$x_{36} = 15.9025787613332$$
$$x_{37} = -83.972087632902$$
$$x_{38} = -97.7343881376139$$
$$x_{39} = -93.881522316429$$
$$x_{40} = -77.8367964156153$$
$$x_{41} = -75.8125376133442$$
$$x_{42} = -31.8794194517317$$
$$x_{43} = -7.7808229418531$$
$$x_{44} = 99.0596546447822$$
Maxima of the function at points:
$$x_{44} = -57.6114316103265$$
$$x_{44} = 66.0699800161888$$
$$x_{44} = 56.175913886337$$
$$x_{44} = 29.829741482418$$
$$x_{44} = 78.1389021841115$$
$$x_{44} = -36.1766711135295$$
$$x_{44} = -14.0032317322368$$
$$x_{44} = -42.0743813758713$$
$$x_{44} = 64.3994668189113$$
$$x_{44} = -29.3120570723519$$
$$x_{44} = 84.2894731854299$$
$$x_{44} = -42.4203679035566$$
$$x_{44} = -71.8554832810555$$
$$x_{44} = -16.0009491904737$$
$$x_{44} = -53.7465455610153$$
$$x_{44} = -62.0232911735757$$
$$x_{44} = 51.8487347950227$$
$$x_{44} = -62.8784063418455$$
$$x_{44} = 4.70091565676332$$
$$x_{44} = 10.2562149360732$$
$$x_{44} = 82.461938457206$$
$$x_{44} = 60.2503535867024$$
$$x_{44} = -26.4373808099425$$
$$x_{44} = 61.1809301963892$$
$$x_{44} = -33.7001911481362$$
$$x_{44} = -51.7503243924615$$
$$x_{44} = 94.8966350594679$$
$$x_{44} = -71.0200484194458$$
$$x_{44} = -47.9381265468237$$
$$x_{44} = -44.0803123955744$$
$$x_{44} = -85.8039257747406$$
$$x_{44} = -67.5251887322074$$
$$x_{44} = 51.8088129558038$$
$$x_{44} = 39.3763165001392$$
$$x_{44} = 73.8280180700468$$
$$x_{44} = 4.13465262627828$$
$$x_{44} = -8.22787376837475$$
$$x_{44} = 47.7411566321408$$
$$x_{44} = -82.1183644934327$$
$$x_{44} = 6.26667579623329$$
$$x_{44} = 2.23412500791198$$
$$x_{44} = -19.9339927192786$$
$$x_{44} = -90.2995543164443$$
$$x_{44} = -59.6740765195264$$
$$x_{44} = 67.5251887322074$$
$$x_{44} = 92.634412032881$$
$$x_{44} = -55.8393982203108$$
$$x_{44} = -81.8885110144078$$
$$x_{44} = 86.4081996571392$$
$$x_{44} = -73.7952790938783$$
$$x_{44} = 37.536735170322$$
$$x_{44} = -17.2772507335565$$
$$x_{44} = -5.7478922734498$$
$$x_{44} = -17.7702721697402$$
$$x_{44} = 26.7424722578364$$
Decreasing at intervals
$$\left[99.0596546447822, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -98.8528049848859\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$2 x \cos{\left(x \right)} \cos{\left(x^{2} \right)} - \sin{\left(x \right)} \sin{\left(x^{2} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -21.7443208246461$$
$$x_{2} = -57.6114316103265$$
$$x_{3} = 66.0699800161888$$
$$x_{4} = 56.175913886337$$
$$x_{5} = -54.2410486649869$$
$$x_{6} = 20.4309423892029$$
$$x_{7} = -36.0818579563209$$
$$x_{8} = 29.829741482418$$
$$x_{9} = 82.7850976847909$$
$$x_{10} = 78.1389021841115$$
$$x_{11} = -10.957767460846$$
$$x_{12} = 62.2002992999418$$
$$x_{13} = -36.1766711135295$$
$$x_{14} = 98.967318895804$$
$$x_{15} = -14.0032317322368$$
$$x_{16} = -42.0743813758713$$
$$x_{17} = 46.032869960288$$
$$x_{18} = 64.3994668189113$$
$$x_{19} = 26.7041238409344$$
$$x_{20} = -29.3120570723519$$
$$x_{21} = 22.3144555829732$$
$$x_{22} = 84.2894731854299$$
$$x_{23} = -42.4203679035566$$
$$x_{24} = -71.8554832810555$$
$$x_{25} = 42.1860239023499$$
$$x_{26} = -16.0009491904737$$
$$x_{27} = -53.7465455610153$$
$$x_{28} = -67.6216177692932$$
$$x_{29} = -62.0232911735757$$
$$x_{30} = 34.2088961514732$$
$$x_{31} = 51.8487347950227$$
$$x_{32} = -79.8291866176748$$
$$x_{33} = -23.749242427506$$
$$x_{34} = 7.7808229418531$$
$$x_{35} = -62.8784063418455$$
$$x_{36} = 4.70091565676332$$
$$x_{37} = 26.259086539065$$
$$x_{38} = 10.2562149360732$$
$$x_{39} = 36.1408978109486$$
$$x_{40} = -3.74766067133477$$
$$x_{41} = 82.461938457206$$
$$x_{42} = 60.2503535867024$$
$$x_{43} = -26.4373808099425$$
$$x_{44} = 61.1809301963892$$
$$x_{45} = 18.0328686012482$$
$$x_{46} = -65.8556721673776$$
$$x_{47} = -33.7001911481362$$
$$x_{48} = -51.7503243924615$$
$$x_{49} = 94.8966350594679$$
$$x_{50} = 86.7145856729587$$
$$x_{51} = 96.0811603317475$$
$$x_{52} = 23.6214725471253$$
$$x_{53} = 0$$
$$x_{54} = 90.3169469354969$$
$$x_{55} = -92.8385290764838$$
$$x_{56} = -71.0200484194458$$
$$x_{57} = -47.9381265468237$$
$$x_{58} = -45.5148417861163$$
$$x_{59} = -44.0803123955744$$
$$x_{60} = -85.8039257747406$$
$$x_{61} = -1.67484432484861$$
$$x_{62} = 54.9762522924857$$
$$x_{63} = -67.5251887322074$$
$$x_{64} = 51.8088129558038$$
$$x_{65} = -95.8526961740184$$
$$x_{66} = -98.8528049848859$$
$$x_{67} = 76.4315416810569$$
$$x_{68} = 39.3763165001392$$
$$x_{69} = 73.8280180700468$$
$$x_{70} = 4.13465262627828$$
$$x_{71} = 14.2458403046005$$
$$x_{72} = 15.9025787613332$$
$$x_{73} = -83.972087632902$$
$$x_{74} = -8.22787376837475$$
$$x_{75} = -97.7343881376139$$
$$x_{76} = -93.881522316429$$
$$x_{77} = 47.7411566321408$$
$$x_{78} = -77.8367964156153$$
$$x_{79} = -75.8125376133442$$
$$x_{80} = -82.1183644934327$$
$$x_{81} = 6.26667579623329$$
$$x_{82} = 2.23412500791198$$
$$x_{83} = -19.9339927192786$$
$$x_{84} = -31.8794194517317$$
$$x_{85} = -90.2995543164443$$
$$x_{86} = -7.7808229418531$$
$$x_{87} = 99.0596546447822$$
$$x_{88} = -59.6740765195264$$
$$x_{89} = 67.5251887322074$$
$$x_{90} = 92.634412032881$$
$$x_{91} = -55.8393982203108$$
$$x_{92} = -81.8885110144078$$
$$x_{93} = 86.4081996571392$$
$$x_{94} = -73.7952790938783$$
$$x_{95} = 37.536735170322$$
$$x_{96} = -17.2772507335565$$
$$x_{97} = -5.7478922734498$$
$$x_{98} = -17.7702721697402$$
$$x_{99} = 26.7424722578364$$
The values of the extrema at the points:
(-21.744320824646053, -0.969676097221814)
(-57.611431610326534, 0.486400074331572)
(66.0699800161888, 0.995343915349401)
(56.175913886337, 0.931322325891384)
(-54.24104866498691, -0.671903550807613)
(20.430942389202887, -0.00420586068706861)
(-36.081857956320945, -0.0445058898446669)
(29.829741482418022, 0.0104073009335254)
(82.78509768479086, -0.450273366645496)
(78.13890218411152, 0.920701233646281)
(-10.95776746084597, -0.0241222477221928)
(62.20029929994178, -0.80709718360807)
(-36.17667111352951, 0.046478368154231)
(98.96731889580397, -0.00583967022909107)
(-14.003231732236845, 0.129079844864294)
(-42.074381375871305, 0.330580102335346)
(46.032869960288, -0.461480368412829)
(64.39946681891129, 0.00120710383472745)
(26.704123840934372, -1.8349056647797e-5)
(-29.312057072351926, 0.507970255345382)
(22.314455582973224, -0.948163233336424)
(84.28947318542987, 0.861012663162526)
(-42.42036790355657, 0.00533064495466314)
(-71.85548328105554, 0.92060942237565)
(42.18602390234994, -0.223273371835521)
(-16.00094919047374, 0.957343240124375)
(-53.74654556101527, 0.942926043018534)
(-67.62161776929321, -0.0769493800481582)
(-62.02329117357569, 0.690514702163622)
(34.20889615147322, -0.93983071729092)
(51.84873479502273, 0.00984917104535034)
(-79.82918661767485, -0.27766072270618)
(-23.749242427506, -0.185065957824233)
(7.780822941853101, -0.0549592504575931)
(-62.87840634184554, 0.998916523665929)
(4.700915656763321, 0.00123051841008508)
(26.25908653906497, -0.42961903305635)
(10.256214936073231, 0.672853864692745)
(36.14089781094858, -0.00846572461553774)
(-3.7476606713347747, -0.818402772365498)
(82.46193845720597, 0.7105282681268)
(60.25035358670237, 0.847194146133567)
(-26.437380809942546, 0.262394827342632)
(61.180930196389205, 0.0796294820628317)
(18.032868601248204, -0.684341420447224)
(-65.85567216737758, -0.993072306512991)
(-33.700191148136184, 0.654363939152423)
(-51.75032439246152, 0.0853139441750081)
(94.89663505946787, 0.796769580838193)
(86.71458567295866, -0.315266706524861)
(96.08116033174753, -0.259528591129436)
(23.62147254712529, -0.0560615995086132)
(0, 0)
(90.31694693549686, -0.704374006726898)
(-92.83852907648381, -0.160756304643403)
(-71.02004841944581, 0.327959090743256)
(-47.938126546823675, 0.686381734150551)
(-45.514841786116335, -0.0367579709104531)
(-44.080312395574445, 0.995199731182475)
(-85.8039257747406, 0.556233746167078)
(-1.674844324848613, -0.0342921123704619)
(54.976252292485654, -0.000283792846225007)
(-67.52518873220743, 0.0177585582888914)
(51.808812955803766, 0.0259101624261079)
(-95.8526961740184, -0.0337221111406507)
(-98.85280498488589, -0.10703965101645)
(76.4315416810569, -0.511940665927364)
(39.37631650013918, 0.10546498949545)
(73.82801807004678, 5.1328095717406e-5)
(4.134652626278284, 0.5369693932868)
(14.245840304600469, -0.103248603871224)
(15.902578761333166, -0.981103260528867)
(-83.97208763290197, -0.659280989408808)
(-8.22787376837475, 0.360937712244325)
(-97.73438813761395, -0.941068492535655)
(-93.88152231642897, -0.933672285597302)
(47.741156632140836, 0.815440953086963)
(-77.83679641561532, -0.762881909929572)
(-75.81253761334425, -0.915388872550898)
(-82.11836449343274, 0.906040577000121)
(6.266675796233288, 0.999862853512404)
(2.234125007911981, 0.591945725863939)
(-19.933992719278617, 0.466885578466679)
(-31.879419451731668, -0.894468878964384)
(-90.29955431644433, 0.691921908953395)
(-7.780822941853101, -0.0549592504575931)
(99.05965464478224, -0.0991952820278843)
(-59.6740765195264, 0.999869034375718)
(67.52518873220743, 0.0177585582888914)
(92.63441203288097, 0.0422207914789287)
(-55.839398220310784, 0.758815401718861)
(-81.8885110144078, 0.978630114019052)
(86.40819965713925, 0.0133630181218172)
(-73.79527909387828, 0.0314522513610657)
(37.536735170321954, 0.986843498791729)
(-17.27725073355649, 7.85622449909438e-5)
(-5.747892273449797, 0.85897692575953)
(-17.770272169740206, 0.471309393582847)
(26.74247225783638, 0.0350921057346341)
Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = -21.7443208246461$$
$$x_{2} = -54.2410486649869$$
$$x_{3} = 20.4309423892029$$
$$x_{4} = -36.0818579563209$$
$$x_{5} = 82.7850976847909$$
$$x_{6} = -10.957767460846$$
$$x_{7} = 62.2002992999418$$
$$x_{8} = 98.967318895804$$
$$x_{9} = 46.032869960288$$
$$x_{10} = 26.7041238409344$$
$$x_{11} = 22.3144555829732$$
$$x_{12} = 42.1860239023499$$
$$x_{13} = -67.6216177692932$$
$$x_{14} = 34.2088961514732$$
$$x_{15} = -79.8291866176748$$
$$x_{16} = -23.749242427506$$
$$x_{17} = 7.7808229418531$$
$$x_{18} = 26.259086539065$$
$$x_{19} = 36.1408978109486$$
$$x_{20} = -3.74766067133477$$
$$x_{21} = 18.0328686012482$$
$$x_{22} = -65.8556721673776$$
$$x_{23} = 86.7145856729587$$
$$x_{24} = 96.0811603317475$$
$$x_{25} = 23.6214725471253$$
$$x_{26} = 0$$
$$x_{27} = 90.3169469354969$$
$$x_{28} = -92.8385290764838$$
$$x_{29} = -45.5148417861163$$
$$x_{30} = -1.67484432484861$$
$$x_{31} = 54.9762522924857$$
$$x_{32} = -95.8526961740184$$
$$x_{33} = -98.8528049848859$$
$$x_{34} = 76.4315416810569$$
$$x_{35} = 14.2458403046005$$
$$x_{36} = 15.9025787613332$$
$$x_{37} = -83.972087632902$$
$$x_{38} = -97.7343881376139$$
$$x_{39} = -93.881522316429$$
$$x_{40} = -77.8367964156153$$
$$x_{41} = -75.8125376133442$$
$$x_{42} = -31.8794194517317$$
$$x_{43} = -7.7808229418531$$
$$x_{44} = 99.0596546447822$$
Maxima of the function at points:
$$x_{44} = -57.6114316103265$$
$$x_{44} = 66.0699800161888$$
$$x_{44} = 56.175913886337$$
$$x_{44} = 29.829741482418$$
$$x_{44} = 78.1389021841115$$
$$x_{44} = -36.1766711135295$$
$$x_{44} = -14.0032317322368$$
$$x_{44} = -42.0743813758713$$
$$x_{44} = 64.3994668189113$$
$$x_{44} = -29.3120570723519$$
$$x_{44} = 84.2894731854299$$
$$x_{44} = -42.4203679035566$$
$$x_{44} = -71.8554832810555$$
$$x_{44} = -16.0009491904737$$
$$x_{44} = -53.7465455610153$$
$$x_{44} = -62.0232911735757$$
$$x_{44} = 51.8487347950227$$
$$x_{44} = -62.8784063418455$$
$$x_{44} = 4.70091565676332$$
$$x_{44} = 10.2562149360732$$
$$x_{44} = 82.461938457206$$
$$x_{44} = 60.2503535867024$$
$$x_{44} = -26.4373808099425$$
$$x_{44} = 61.1809301963892$$
$$x_{44} = -33.7001911481362$$
$$x_{44} = -51.7503243924615$$
$$x_{44} = 94.8966350594679$$
$$x_{44} = -71.0200484194458$$
$$x_{44} = -47.9381265468237$$
$$x_{44} = -44.0803123955744$$
$$x_{44} = -85.8039257747406$$
$$x_{44} = -67.5251887322074$$
$$x_{44} = 51.8088129558038$$
$$x_{44} = 39.3763165001392$$
$$x_{44} = 73.8280180700468$$
$$x_{44} = 4.13465262627828$$
$$x_{44} = -8.22787376837475$$
$$x_{44} = 47.7411566321408$$
$$x_{44} = -82.1183644934327$$
$$x_{44} = 6.26667579623329$$
$$x_{44} = 2.23412500791198$$
$$x_{44} = -19.9339927192786$$
$$x_{44} = -90.2995543164443$$
$$x_{44} = -59.6740765195264$$
$$x_{44} = 67.5251887322074$$
$$x_{44} = 92.634412032881$$
$$x_{44} = -55.8393982203108$$
$$x_{44} = -81.8885110144078$$
$$x_{44} = 86.4081996571392$$
$$x_{44} = -73.7952790938783$$
$$x_{44} = 37.536735170322$$
$$x_{44} = -17.2772507335565$$
$$x_{44} = -5.7478922734498$$
$$x_{44} = -17.7702721697402$$
$$x_{44} = 26.7424722578364$$
Decreasing at intervals
$$\left[99.0596546447822, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -98.8528049848859\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(x^{2} \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(x^{2} \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(x^{2} \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(x^{2} \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)*sin(x^2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} \right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} \right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\sin{\left(x^{2} \right)} \cos{\left(x \right)} = \sin{\left(x^{2} \right)} \cos{\left(x \right)}$$
- Yes
$$\sin{\left(x^{2} \right)} \cos{\left(x \right)} = - \sin{\left(x^{2} \right)} \cos{\left(x \right)}$$
- No
so, the function
is
even
So, check:
$$\sin{\left(x^{2} \right)} \cos{\left(x \right)} = \sin{\left(x^{2} \right)} \cos{\left(x \right)}$$
- Yes
$$\sin{\left(x^{2} \right)} \cos{\left(x \right)} = - \sin{\left(x^{2} \right)} \cos{\left(x \right)}$$
- No
so, the function
is
even