Графики функций/ Построение в 2D/ y = (-cos(x))*cos(5*x)

График функции y = (-cos(x))*cos(5*x)

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Примеры
v

График:

от до

Точки пересечения:

Кусочно-заданная:

{ кусочно-заданную функцию ввести здесь.

Решение

Вы ввели [src]
f(x) = -cos(x)*cos(5*x)
f(x)=−cos⁡(x)cos⁡(5x)f{\left (x \right )} = - \cos{\left (x \right )} \cos{\left (5 x \right )}
График функции
0-2000-1500-1000-5005001000150020002-2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0
значит надо решить уравнение:
−cos⁡(x)cos⁡(5x)=0- \cos{\left (x \right )} \cos{\left (5 x \right )} = 0
Решаем это уравнение
Точки пересечения с осью X:

Аналитическое решение
x1=π10x_{1} = \frac{\pi}{10}
x2=3π10x_{2} = \frac{3 \pi}{10}
x3=π2x_{3} = \frac{\pi}{2}
x4=3π2x_{4} = \frac{3 \pi}{2}
Численное решение
x1=−83.8805238508x_{1} = -83.8805238508
x2=−61.8893752757x_{2} = -61.8893752757
x3=70.0575161751x_{3} = 70.0575161751
x4=88.2787535659x_{4} = 88.2787535659
x5=42.4115007836x_{5} = 42.4115007836
x6=−73.8274272837x_{6} = -73.8274272837
x7=−51.8362786978x_{7} = -51.8362786978
x8=−27.9601746169x_{8} = -27.9601746169
x9=78.2256570744x_{9} = 78.2256570744
x10=32.358404332x_{10} = 32.358404332
x11=−75.7123829515x_{11} = -75.7123829515
x12=−38.0132711084x_{12} = -38.0132711084
x13=54.3495529071x_{13} = 54.3495529071
x14=100.21680565x_{14} = 100.21680565
x15=81.9955682587x_{15} = 81.9955682587
x16=12.252211349x_{16} = 12.252211349
x17=−97.7035315266x_{17} = -97.7035315266
x18=95.818575945x_{18} = 95.818575945
x19=−29.8451301142x_{19} = -29.8451301142
x20=22.3053078405x_{20} = 22.3053078405
x21=39.2699081498x_{21} = 39.2699081498
x22=29.8451302545x_{22} = 29.8451302545
x23=−80.1106126089x_{23} = -80.1106126089
x24=−55.6061899685x_{24} = -55.6061899685
x25=−43.6681378849x_{25} = -43.6681378849
x26=−58.1194640341x_{26} = -58.1194640341
x27=5.96902604182x_{27} = 5.96902604182
x28=73.8274273864x_{28} = 73.8274273864
x29=−39.8982267006x_{29} = -39.8982267006
x30=86.3937979238x_{30} = 86.3937979238
x31=−17.9070781255x_{31} = -17.9070781255
x32=−71.9424717672x_{32} = -71.9424717672
x33=2.19911485751x_{33} = 2.19911485751
x34=61.8893752757x_{34} = 61.8893752757
x35=−36.1283154602x_{35} = -36.1283154602
x36=20.4203522182x_{36} = 20.4203522182
x37=−48.0663675999x_{37} = -48.0663675999
x38=64.4026493525x_{38} = 64.4026493525
x39=−53.7212343764x_{39} = -53.7212343764
x40=−11.6238928183x_{40} = -11.6238928183
x41=−9.73893722613x_{41} = -9.73893722613
x42=17.9070781255x_{42} = 17.9070781255
x43=7.85398168447x_{43} = 7.85398168447
x44=48.0663675999x_{44} = 48.0663675999
x45=71.9424717672x_{45} = 71.9424717672
x46=−45.5530934329x_{46} = -45.5530934329
x47=10.3672557568x_{47} = 10.3672557568
x48=−89.5353904623x_{48} = -89.5353904623
x49=−16.0221225333x_{49} = -16.0221225333
x50=−60.0044196836x_{50} = -60.0044196836
x51=−33.6150413934x_{51} = -33.6150413934
x52=92.0486647502x_{52} = 92.0486647502
x53=58.7477826221x_{53} = 58.7477826221
x54=24.1902634326x_{54} = 24.1902634326
x55=4.08407044967x_{55} = 4.08407044967
x56=−7.85398154718x_{56} = -7.85398154718
x57=76.3407014822x_{57} = 76.3407014822
x58=66.2876049907x_{58} = 66.2876049907
x59=−63.7743308679x_{59} = -63.7743308679
x60=90.163709158x_{60} = 90.163709158
x61=93.9336203423x_{61} = 93.9336203423
x62=−19.7920337176x_{62} = -19.7920337176
x63=−1.57079634218x_{63} = -1.57079634218
x64=46.1814120078x_{64} = 46.1814120078
x65=44.2964564156x_{65} = 44.2964564156
x66=51.8362788222x_{66} = 51.8362788222
x67=−70.0575161751x_{67} = -70.0575161751
x68=−21.6769893098x_{68} = -21.6769893098
x69=34.2433599241x_{69} = 34.2433599241
x70=27.9601746169x_{70} = 27.9601746169
x71=−31.7300858013x_{71} = -31.7300858013
x72=60.0044196836x_{72} = 60.0044196836
x73=−49.9513231921x_{73} = -49.9513231921
x74=83.8805238508x_{74} = 83.8805238508
x75=56.2345084993x_{75} = 56.2345084993
x76=−77.5973385437x_{76} = -77.5973385437
x77=−4.08407044967x_{77} = -4.08407044967
x78=−67.5442419493x_{78} = -67.5442419493
x79=−5.96902604182x_{79} = -5.96902604182
x80=−65.65928646x_{80} = -65.65928646
x81=−93.9336203423x_{81} = -93.9336203423
x82=−23.5619448943x_{82} = -23.5619448943
x83=49.9513231921x_{83} = 49.9513231921
x84=68.1725605829x_{84} = 68.1725605829
x85=−14.1371668872x_{85} = -14.1371668872
x86=16.0221225333x_{86} = 16.0221225333
x87=0.314159265359x_{87} = 0.314159265359
x88=−87.6504350352x_{88} = -87.6504350352
x89=−41.7831822927x_{89} = -41.7831822927
x90=26.0752190248x_{90} = 26.0752190248
x91=−92.0486647502x_{91} = -92.0486647502
x92=98.3318500574x_{92} = 98.3318500574
x93=−81.9955682587x_{93} = -81.9955682587
x94=−85.765479443x_{94} = -85.765479443
x95=−95.8185758696x_{95} = -95.8185758696
x96=38.0132711084x_{96} = 38.0132711084
x97=39.8982267006x_{97} = 39.8982267006
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:
подставляем x = 0 в (-cos(x))*cos(5*x).
−cos⁡(0)cos⁡(0⋅5)- \cos{\left (0 \right )} \cos{\left (0 \cdot 5 \right )}
Результат:
f(0)=−1f{\left (0 \right )} = -1
Точка:
(0, -1)
Экстремумы функции
Для того, чтобы найти экстремумы, нужно решить уравнение
ddxf(x)=0\frac{d}{d x} f{\left (x \right )} = 0
(производная равна нулю),
и корни этого уравнения будут экстремумами данной функции:
ddxf(x)=\frac{d}{d x} f{\left (x \right )} =
Первая производная
sin⁡(x)cos⁡(5x)+5sin⁡(5x)cos⁡(x)=0\sin{\left (x \right )} \cos{\left (5 x \right )} + 5 \sin{\left (5 x \right )} \cos{\left (x \right )} = 0
Решаем это уравнение
Корни этого ур-ния
x1=−71.6555702774x_{1} = -71.6555702774
x2=−93.6467188526x_{2} = -93.6467188526
x3=−95.8185759345x_{3} = -95.8185759345
x4=10.0258387159x_{4} = 10.0258387159
x5=92.2749008201x_{5} = 92.2749008201
x6=−3.74265340872x_{6} = -3.74265340872
x7=74.2295098202x_{7} = 74.2295098202
x8=−85.9917155129x_{8} = -85.9917155129
x9=99.9299041597x_{9} = 99.9299041597
x10=−39.6719906307x_{10} = -39.6719906307
x11=30.2472126699x_{11} = 30.2472126699
x12=36.1283155163x_{12} = 36.1283155163
x13=−77.9387555846x_{13} = -77.9387555846
x14=80.1106126665x_{14} = 80.1106126665
x15=−20.0182697875x_{15} = -20.0182697875
x16=62.2307923167x_{16} = 62.2307923167
x17=64.0005669378x_{17} = 64.0005669378
x18=14.1371669412x_{18} = 14.1371669412
x19=−5.68212455205x_{19} = -5.68212455205
x20=−83.6542877809x_{20} = -83.6542877809
x21=−89.9374730881x_{21} = -89.9374730881
x22=−51.8362787842x_{22} = -51.8362787842
x23=97.9904330164x_{23} = 97.9904330164
x24=−69.7160991341x_{24} = -69.7160991341
x25=11.9653098592x_{25} = 11.9653098592
x26=−49.6644217023x_{26} = -49.6644217023
x27=−75.9992844413x_{27} = -75.9992844413
x28=8.25606409479x_{28} = 8.25606409479
x29=43.9822971503x_{29} = 43.9822971503
x30=−11.9653098592x_{30} = -11.9653098592
x31=−10.0258387159x_{31} = -10.0258387159
x32=−7.85398163397x_{32} = -7.85398163397
x33=75.9992844413x_{33} = 75.9992844413
x34=33.9564584344x_{34} = 33.9564584344
x35=−23.9640273627x_{35} = -23.9640273627
x36=−57.7173816306x_{36} = -57.7173816306
x37=55.9476070095x_{37} = 55.9476070095
x38=−1.97287878761x_{38} = -1.97287878761
x39=−17.6808420556x_{39} = -17.6808420556
x40=87.9645943005x_{40} = 87.9645943005
x41=−27.6732731272x_{41} = -27.6732731272
x42=26.3014550947x_{42} = 26.3014550947
x43=38.3001725982x_{43} = 38.3001725982
x44=−32.016987291x_{44} = -32.016987291
x45=84.2219408918x_{45} = 84.2219408918
x46=−21.9911485751x_{46} = -21.9911485751
x47=−37.6991118431x_{47} = -37.6991118431
x48=54.0081358662x_{48} = 54.0081358662
x49=21.9911485751x_{49} = 21.9911485751
x50=−29.8451302091x_{50} = -29.8451302091
x51=0x_{51} = 0
x52=4.31030651957x_{52} = 4.31030651957
x53=82.2824697485x_{53} = 82.2824697485
x54=−79.7085302057x_{54} = -79.7085302057
x55=60.2913211733x_{55} = 60.2913211733
x56=−55.9476070095x_{56} = -55.9476070095
x57=50.2654824574x_{57} = 50.2654824574
x58=−59.6902604182x_{58} = -59.6902604182
x59=28.2743338823x_{59} = 28.2743338823
x60=−43.9822971503x_{60} = -43.9822971503
x61=−81.6814089933x_{61} = -81.6814089933
x62=16.3090240231x_{62} = 16.3090240231
x63=72.2566310326x_{63} = 72.2566310326
x64=−65.9734457254x_{64} = -65.9734457254
x65=−54.0081358662x_{65} = -54.0081358662
x66=40.2396437415x_{66} = 40.2396437415
x67=−73.8274273594x_{67} = -73.8274273594
x68=−61.6631392058x_{68} = -61.6631392058
x69=−99.9299041597x_{69} = -99.9299041597
x70=−35.7262330555x_{70} = -35.7262330555
x71=70.283752245x_{71} = 70.283752245
x72=32.016987291x_{72} = 32.016987291
x73=94.2477796077x_{73} = 94.2477796077
x74=96.2206583953x_{74} = 96.2206583953
x75=89.9374730881x_{75} = 89.9374730881
x76=−97.9904330164x_{76} = -97.9904330164
x77=−15.7079632679x_{77} = -15.7079632679
x78=42.0094183626x_{78} = 42.0094183626
x79=48.2926036698x_{79} = 48.2926036698
x80=−33.9564584344x_{80} = -33.9564584344
x81=−47.724950559x_{81} = -47.724950559
x82=65.9734457254x_{82} = 65.9734457254
x83=85.9917155129x_{83} = 85.9917155129
x84=−13.7350844803x_{84} = -13.7350844803
x85=52.238361245x_{85} = 52.238361245
x86=67.946324513x_{86} = 67.946324513
x87=58.1194640914x_{87} = 58.1194640914
x88=18.2484951664x_{88} = 18.2484951664
x89=1.97287878761x_{89} = 1.97287878761
x90=77.9387555846x_{90} = 77.9387555846
x91=−42.0094183626x_{91} = -42.0094183626
x92=−91.7072477092x_{92} = -91.7072477092
x93=6.28318530718x_{93} = 6.28318530718
x94=−67.946324513x_{94} = -67.946324513
x95=−87.9645943005x_{95} = -87.9645943005
x96=45.9551759379x_{96} = 45.9551759379
x97=20.0182697875x_{97} = 20.0182697875
x98=−25.7338019838x_{98} = -25.7338019838
x99=−45.9551759379x_{99} = -45.9551759379
x100=−64.0005669378x_{100} = -64.0005669378
x101=23.9640273627x_{101} = 23.9640273627
Зн. экстремумы в точках:
(-71.6555702774, 0.817088455586739)

(-93.6467188526, 0.81708845558674)

(-95.8185759345, -6.39727825033797e-22)

(10.0258387159, 0.817088455586737)

(92.2749008201, -0.354125492623776)

(-3.74265340872, 0.817088455586737)

(74.2295098202, -0.354125492623772)

(-85.9917155129, -0.354125492623772)

(99.9299041597, 0.817088455586735)

(-39.6719906307, -0.354125492623773)

(30.2472126699, -0.354125492623775)

(36.1283155163, -1.50987035263964e-21)

(-77.9387555846, 0.817088455586735)

(80.1106126665, -7.89007547709721e-21)

(-20.0182697875, -0.354125492623775)

(62.2307923167, 0.817088455586734)

(64.0005669378, -0.354125492623772)

(14.1371669412, -1.05484759088614e-20)

(-5.68212455205, 0.817088455586737)

(-83.6542877809, -0.354125492623774)

(-89.9374730881, -0.354125492623779)

(-51.8362787842, -4.98972383209749e-21)

(97.9904330164, 0.817088455586737)

(-69.7160991341, 0.817088455586734)

(11.9653098592, 0.817088455586737)

(-49.6644217023, 0.817088455586738)

(-75.9992844413, 0.817088455586737)

(8.25606409479, -0.354125492623774)

(43.9822971503, -1)

(-11.9653098592, 0.817088455586737)

(-10.0258387159, 0.817088455586737)

(-7.85398163397, -1.00495818792069e-22)

(75.9992844413, 0.817088455586737)

(33.9564584344, 0.817088455586738)

(-23.9640273627, -0.354125492623775)

(-57.7173816306, -0.354125492623776)

(55.9476070095, 0.817088455586738)

(-1.97287878761, -0.354125492623774)

(-17.6808420556, -0.354125492623775)

(87.9645943005, -1)

(-27.6732731272, 0.817088455586737)

(26.3014550947, -0.354125492623772)

(38.3001725982, 0.817088455586736)

(-32.016987291, 0.817088455586736)

(84.2219408918, 0.817088455586738)

(-21.9911485751, -1)

(-37.6991118431, -1)

(54.0081358662, 0.817088455586734)

(21.9911485751, -1)

(-29.8451302091, -4.60939679034216e-23)

(0, -1)

(4.31030651957, -0.354125492623774)

(82.2824697485, 0.81708845558674)

(-79.7085302057, -0.354125492623776)

(60.2913211733, 0.817088455586739)

(-55.9476070095, 0.817088455586738)

(50.2654824574, -1)

(-59.6902604182, -1)

(28.2743338823, -1)

(-43.9822971503, -1)

(-81.6814089933, -1)

(16.3090240231, 0.817088455586736)

(72.2566310326, -1)

(-65.9734457254, -1)

(-54.0081358662, 0.817088455586734)

(40.2396437415, 0.817088455586736)

(-73.8274273594, -7.94621856866359e-21)

(-61.6631392058, -0.354125492623778)

(-99.9299041597, 0.817088455586735)

(-35.7262330555, -0.354125492623776)

(70.283752245, -0.354125492623776)

(32.016987291, 0.817088455586736)

(94.2477796077, -1)

(96.2206583953, -0.354125492623774)

(89.9374730881, -0.354125492623779)

(-97.9904330164, 0.817088455586737)

(-15.7079632679, -1)

(42.0094183626, -0.354125492623774)

(48.2926036698, -0.354125492623774)

(-33.9564584344, 0.817088455586738)

(-47.724950559, 0.817088455586738)

(65.9734457254, -1)

(85.9917155129, -0.354125492623772)

(-13.7350844803, -0.354125492623775)

(52.238361245, -0.354125492623775)

(67.946324513, -0.354125492623776)

(58.1194640914, -6.24901394491486e-22)

(18.2484951664, 0.817088455586736)

(1.97287878761, -0.354125492623774)

(77.9387555846, 0.817088455586735)

(-42.0094183626, -0.354125492623774)

(-91.7072477092, 0.81708845558674)

(6.28318530718, -1)

(-67.946324513, -0.354125492623776)

(-87.9645943005, -1)

(45.9551759379, -0.354125492623773)

(20.0182697875, -0.354125492623775)

(-25.7338019838, 0.817088455586736)

(-45.9551759379, -0.354125492623773)

(-64.0005669378, -0.354125492623772)

(23.9640273627, -0.354125492623775)


Интервалы возрастания и убывания функции:
Найдём интервалы, где функция возрастает и убывает, а также минимумы и максимумы функции, для этого смотрим как ведёт себя функция в экстремумах при малейшем отклонении от экстремума:
Минимумы функции в точках:
x101=92.2749008201x_{101} = 92.2749008201
x101=74.2295098202x_{101} = 74.2295098202
x101=−85.9917155129x_{101} = -85.9917155129
x101=−39.6719906307x_{101} = -39.6719906307
x101=30.2472126699x_{101} = 30.2472126699
x101=−20.0182697875x_{101} = -20.0182697875
x101=64.0005669378x_{101} = 64.0005669378
x101=−83.6542877809x_{101} = -83.6542877809
x101=−89.9374730881x_{101} = -89.9374730881
x101=8.25606409479x_{101} = 8.25606409479
x101=43.9822971503x_{101} = 43.9822971503
x101=−23.9640273627x_{101} = -23.9640273627
x101=−57.7173816306x_{101} = -57.7173816306
x101=−1.97287878761x_{101} = -1.97287878761
x101=−17.6808420556x_{101} = -17.6808420556
x101=87.9645943005x_{101} = 87.9645943005
x101=26.3014550947x_{101} = 26.3014550947
x101=−21.9911485751x_{101} = -21.9911485751
x101=−37.6991118431x_{101} = -37.6991118431
x101=21.9911485751x_{101} = 21.9911485751
x101=0x_{101} = 0
x101=4.31030651957x_{101} = 4.31030651957
x101=−79.7085302057x_{101} = -79.7085302057
x101=50.2654824574x_{101} = 50.2654824574
x101=−59.6902604182x_{101} = -59.6902604182
x101=28.2743338823x_{101} = 28.2743338823
x101=−43.9822971503x_{101} = -43.9822971503
x101=−81.6814089933x_{101} = -81.6814089933
x101=72.2566310326x_{101} = 72.2566310326
x101=−65.9734457254x_{101} = -65.9734457254
x101=−61.6631392058x_{101} = -61.6631392058
x101=−35.7262330555x_{101} = -35.7262330555
x101=70.283752245x_{101} = 70.283752245
x101=94.2477796077x_{101} = 94.2477796077
x101=96.2206583953x_{101} = 96.2206583953
x101=89.9374730881x_{101} = 89.9374730881
x101=−15.7079632679x_{101} = -15.7079632679
x101=42.0094183626x_{101} = 42.0094183626
x101=48.2926036698x_{101} = 48.2926036698
x101=65.9734457254x_{101} = 65.9734457254
x101=85.9917155129x_{101} = 85.9917155129
x101=−13.7350844803x_{101} = -13.7350844803
x101=52.238361245x_{101} = 52.238361245
x101=67.946324513x_{101} = 67.946324513
x101=1.97287878761x_{101} = 1.97287878761
x101=−42.0094183626x_{101} = -42.0094183626
x101=6.28318530718x_{101} = 6.28318530718
x101=−67.946324513x_{101} = -67.946324513
x101=−87.9645943005x_{101} = -87.9645943005
x101=45.9551759379x_{101} = 45.9551759379
x101=20.0182697875x_{101} = 20.0182697875
x101=−45.9551759379x_{101} = -45.9551759379
x101=−64.0005669378x_{101} = -64.0005669378
x101=23.9640273627x_{101} = 23.9640273627
Максимумы функции в точках:
x101=−71.6555702774x_{101} = -71.6555702774
x101=−93.6467188526x_{101} = -93.6467188526
x101=−95.8185759345x_{101} = -95.8185759345
x101=10.0258387159x_{101} = 10.0258387159
x101=−3.74265340872x_{101} = -3.74265340872
x101=99.9299041597x_{101} = 99.9299041597
x101=36.1283155163x_{101} = 36.1283155163
x101=−77.9387555846x_{101} = -77.9387555846
x101=80.1106126665x_{101} = 80.1106126665
x101=62.2307923167x_{101} = 62.2307923167
x101=14.1371669412x_{101} = 14.1371669412
x101=−5.68212455205x_{101} = -5.68212455205
x101=−51.8362787842x_{101} = -51.8362787842
x101=97.9904330164x_{101} = 97.9904330164
x101=−69.7160991341x_{101} = -69.7160991341
x101=11.9653098592x_{101} = 11.9653098592
x101=−49.6644217023x_{101} = -49.6644217023
x101=−75.9992844413x_{101} = -75.9992844413
x101=−11.9653098592x_{101} = -11.9653098592
x101=−10.0258387159x_{101} = -10.0258387159
x101=−7.85398163397x_{101} = -7.85398163397
x101=75.9992844413x_{101} = 75.9992844413
x101=33.9564584344x_{101} = 33.9564584344
x101=55.9476070095x_{101} = 55.9476070095
x101=−27.6732731272x_{101} = -27.6732731272
x101=38.3001725982x_{101} = 38.3001725982
x101=−32.016987291x_{101} = -32.016987291
x101=84.2219408918x_{101} = 84.2219408918
x101=54.0081358662x_{101} = 54.0081358662
x101=−29.8451302091x_{101} = -29.8451302091
x101=82.2824697485x_{101} = 82.2824697485
x101=60.2913211733x_{101} = 60.2913211733
x101=−55.9476070095x_{101} = -55.9476070095
x101=16.3090240231x_{101} = 16.3090240231
x101=−54.0081358662x_{101} = -54.0081358662
x101=40.2396437415x_{101} = 40.2396437415
x101=−73.8274273594x_{101} = -73.8274273594
x101=−99.9299041597x_{101} = -99.9299041597
x101=32.016987291x_{101} = 32.016987291
x101=−97.9904330164x_{101} = -97.9904330164
x101=−33.9564584344x_{101} = -33.9564584344
x101=−47.724950559x_{101} = -47.724950559
x101=58.1194640914x_{101} = 58.1194640914
x101=18.2484951664x_{101} = 18.2484951664
x101=77.9387555846x_{101} = 77.9387555846
x101=−91.7072477092x_{101} = -91.7072477092
x101=−25.7338019838x_{101} = -25.7338019838
Убывает на промежутках
[96.2206583953, oo)

Возрастает на промежутках
(-oo, -89.9374730881]
Точки перегибов
Найдем точки перегибов, для этого надо решить уравнение
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left (x \right )} = 0
(вторая производная равняется нулю),
корни полученного уравнения будут точками перегибов для указанного графика функции:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left (x \right )} =
Вторая производная
2(−5sin⁡(x)sin⁡(5x)+13cos⁡(x)cos⁡(5x))=02 \left(- 5 \sin{\left (x \right )} \sin{\left (5 x \right )} + 13 \cos{\left (x \right )} \cos{\left (5 x \right )}\right) = 0
Решаем это уравнение
Корни этого ур-ния
x1=−33.6988683432x_{1} = -33.6988683432
x2=−96.0306615985x_{2} = -96.0306615985
x3=100.23976012x_{3} = 100.23976012
x4=49.9742776629x_{4} = 49.9742776629
x5=−69.9736892253x_{5} = -69.9736892253
x6=−65.6822409308x_{6} = -65.6822409308
x7=−27.9831290877x_{7} = -27.9831290877
x8=−53.6982799056x_{8} = -53.6982799056
x9=−81.9726137879x_{9} = -81.9726137879
x10=69.9736892253x_{10} = 69.9736892253
x11=−39.9820536503x_{11} = -39.9820536503
x12=98.2480231076x_{12} = 98.2480231076
x13=78.2486115452x_{13} = 78.2486115452
x14=−99.6723140685x_{14} = -99.6723140685
x15=15.9991680625x_{15} = 15.9991680625
x16=17.9909050752x_{16} = 17.9909050752
x17=−41.699355343x_{17} = -41.699355343
x18=14.8493124216x_{18} = 14.8493124216
x19=−85.6816524933x_{19} = -85.6816524933
x20=64.1905637345x_{20} = 64.1905637345
x21=0.291204794564x_{21} = 0.291204794564
x22=−63.6905039181x_{22} = -63.6905039181
x23=−9.71598275533x_{23} = -9.71598275533
x24=96.0306615985x_{24} = 96.0306615985
x25=−49.9742776629x_{25} = -49.9742776629
x26=20.2082665843x_{26} = 20.2082665843
x27=−23.774030566x_{27} = -23.774030566
x28=−37.9903166376x_{28} = -37.9903166376
x29=−77.6811654934x_{29} = -77.6811654934
x30=56.2574629701x_{30} = 56.2574629701
x31=−17.9909050752x_{31} = -17.9909050752
x32=30.0572158732x_{32} = 30.0572158732
x33=46.2652389575x_{33} = 46.2652389575
x34=−57.9073784274x_{34} = -57.9073784274
x35=−35.9162298522x_{35} = -35.9162298522
x36=−45.7651791411x_{36} = -45.7651791411
x37=44.2735019448x_{37} = 44.2735019448
x38=−67.7563277162x_{38} = -67.7563277162
x39=−55.6900169183x_{39} = -55.6900169183
x40=2.28294180726x_{40} = 2.28294180726
x41=79.8985270025x_{41} = 79.8985270025
x42=62.5406482772x_{42} = 62.5406482772
x43=71.965426238x_{43} = 71.965426238
x44=−1.78288199086x_{44} = -1.78288199086
x45=−47.9825406502x_{45} = -47.9825406502
x46=47.9825406502x_{46} = 47.9825406502
x47=−11.707719768x_{47} = -11.707719768
x48=54.2657259574x_{48} = 54.2657259574
x49=52.0483644483x_{49} = 52.0483644483
x50=−74.0395130234x_{50} = -74.0395130234
x51=61.9732022255x_{51} = 61.9732022255
x52=74.0395130234x_{52} = 74.0395130234
x53=83.9643508006x_{53} = 83.9643508006
x54=25.9913920751x_{54} = 25.9913920751
x55=37.9903166376x_{55} = 37.9903166376
x56=66.2646505199x_{56} = 66.2646505199
x57=−31.7071313305x_{57} = -31.7071313305
x58=39.9820536503x_{58} = 39.9820536503
x59=−89.7474762914x_{59} = -89.7474762914
x60=5.99198051262x_{60} = 5.99198051262
x61=59.9814652128x_{61} = 59.9814652128
x62=−93.9565748131x_{62} = -93.9565748131
x63=−91.9648378004x_{63} = -91.9648378004
x64=93.9565748131x_{64} = 93.9565748131
x65=4.00024349992x_{65} = 4.00024349992
x66=−21.6999437806x_{66} = -21.6999437806
x67=34.2663143949x_{67} = 34.2663143949
x68=86.1817123097x_{68} = 86.1817123097
x69=10.2834288071x_{69} = 10.2834288071
x70=−30.0572158732x_{70} = -30.0572158732
x71=12.2751658198x_{71} = 12.2751658198
x72=91.9648378004x_{72} = 91.9648378004
x73=68.2563875326x_{73} = 68.2563875326
x74=−83.9643508006x_{74} = -83.9643508006
x75=−4.00024349992x_{75} = -4.00024349992
x76=−15.9991680625x_{76} = -15.9991680625
x77=76.2568745325x_{77} = 76.2568745325
x78=−87.673389506x_{78} = -87.673389506
x79=−13.9250812771x_{79} = -13.9250812771
x80=81.9726137879x_{80} = 81.9726137879
x81=−79.8985270025x_{81} = -79.8985270025
x82=24.2740903824x_{82} = 24.2740903824
x83=−25.9913920751x_{83} = -25.9913920751
x84=−52.0483644483x_{84} = -52.0483644483
x85=−5.99198051262x_{85} = -5.99198051262
x86=32.2745773822x_{86} = 32.2745773822
x87=88.2557990951x_{87} = 88.2557990951
x88=27.9831290877x_{88} = 27.9831290877
x89=−59.9814652128x_{89} = -59.9814652128
x90=42.1994151594x_{90} = 42.1994151594
x91=−71.965426238x_{91} = -71.965426238
x92=8.06606729804x_{92} = 8.06606729804
x93=−61.9732022255x_{93} = -61.9732022255
x94=90.2475361078x_{94} = 90.2475361078
x95=22.2823533697x_{95} = 22.2823533697
x96=−19.7082067679x_{96} = -19.7082067679
x97=−43.6910923557x_{97} = -43.6910923557
x98=−97.6805770558x_{98} = -97.6805770558
x99=−75.6894284807x_{99} = -75.6894284807

Интервалы выпуклости и вогнутости:
Найдём интервалы, где функция выпуклая или вогнутая, для этого посмотрим, как ведет себя функция в точках перегибов:
Вогнутая на промежутках
[100.23976012, oo)

Выпуклая на промежутках
(-oo, -99.6723140685]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oo
lim⁡x→−∞(−cos⁡(x)cos⁡(5x))=⟨−1,1⟩\lim_{x \to -\infty}\left(- \cos{\left (x \right )} \cos{\left (5 x \right )}\right) = \langle -1, 1\rangle
Возьмём предел
значит,
уравнение горизонтальной асимптоты слева:
y=⟨−1,1⟩y = \langle -1, 1\rangle
lim⁡x→∞(−cos⁡(x)cos⁡(5x))=⟨−1,1⟩\lim_{x \to \infty}\left(- \cos{\left (x \right )} \cos{\left (5 x \right )}\right) = \langle -1, 1\rangle
Возьмём предел
значит,
уравнение горизонтальной асимптоты справа:
y=⟨−1,1⟩y = \langle -1, 1\rangle
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции (-cos(x))*cos(5*x), делённой на x при x->+oo и x ->-oo
lim⁡x→−∞(−1xcos⁡(x)cos⁡(5x))=0\lim_{x \to -\infty}\left(- \frac{1}{x} \cos{\left (x \right )} \cos{\left (5 x \right )}\right) = 0
Возьмём предел
значит,
наклонная совпадает с горизонтальной асимптотой справа
lim⁡x→∞(−1xcos⁡(x)cos⁡(5x))=0\lim_{x \to \infty}\left(- \frac{1}{x} \cos{\left (x \right )} \cos{\left (5 x \right )}\right) = 0
Возьмём предел
значит,
наклонная совпадает с горизонтальной асимптотой слева
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).
Итак, проверяем:
−cos⁡(x)cos⁡(5x)=−cos⁡(x)cos⁡(5x)- \cos{\left (x \right )} \cos{\left (5 x \right )} = - \cos{\left (x \right )} \cos{\left (5 x \right )}
- Да
−cos⁡(x)cos⁡(5x)=−−1cos⁡(x)cos⁡(5x)- \cos{\left (x \right )} \cos{\left (5 x \right )} = - -1 \cos{\left (x \right )} \cos{\left (5 x \right )}
- Нет
значит, функция
является
чётной