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$$\cos{\left(x + \frac{\pi}{3} \right)} \operatorname{sign}{\left(\sin{\left(x + \frac{\pi}{3} \right)} \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 9.94837673636768$$
$$x_{2} = -56.025068989018$$
$$x_{3} = 57.0722665402146$$
$$x_{4} = -15.1843644923507$$
$$x_{5} = 60.2138591938044$$
$$x_{6} = 82.2050077689329$$
$$x_{7} = -37.1755130674792$$
$$x_{8} = -52.8834763354282$$
$$x_{9} = 85.3466004225227$$
$$x_{10} = -84.2994028713261$$
$$x_{11} = -87.4409955249159$$
$$x_{12} = 38.2227106186758$$
$$x_{13} = -90.5825881785057$$
$$x_{14} = 66.497044500984$$
$$x_{15} = 31.9395253114962$$
$$x_{16} = 16.2315620435473$$
$$x_{17} = 22.5147473507269$$
$$x_{18} = 28.7979326579064$$
$$x_{19} = -24.60914245312$$
$$x_{20} = 0.523598775598299$$
$$x_{21} = -46.6002910282486$$
$$x_{22} = -78.0162175641465$$
$$x_{23} = 88.4881930761125$$
$$x_{24} = -68.5914396033772$$
$$x_{25} = -8.90117918517108$$
$$x_{26} = 13.0899693899575$$
$$x_{27} = 44.5058959258554$$
$$x_{28} = 79.0634151153431$$
$$x_{29} = -65.4498469497874$$
$$x_{30} = 69.6386371545737$$
$$x_{31} = -442.440965380563$$
$$x_{32} = 19.3731546971371$$
$$x_{33} = -34.0339204138894$$
$$x_{34} = -74.8746249105567$$
$$x_{35} = 91.6297857297023$$
$$x_{36} = -49.7418836818384$$
$$x_{37} = -2.61799387799149$$
$$x_{38} = -81.1578102177363$$
$$x_{39} = 6.80678408277789$$
$$x_{40} = 72.7802298081635$$
$$x_{41} = -43.4586983746588$$
$$x_{42} = -5.75958653158129$$
$$x_{43} = -30.8923277602996$$
$$x_{44} = 35.081117965086$$
$$x_{45} = -40.317105721069$$
$$x_{46} = 97.9129710368819$$
$$x_{47} = -96.8657734856853$$
$$x_{48} = -12.0427718387609$$
$$x_{49} = -62.3082542961976$$
$$x_{50} = -93.7241808320955$$
$$x_{51} = -18.3259571459405$$
$$x_{52} = 101.054563690472$$
$$x_{53} = 94.7713783832921$$
$$x_{54} = 53.9306738866248$$
$$x_{55} = 25.6563400043166$$
$$x_{56} = 3.66519142918809$$
$$x_{57} = -71.733032256967$$
$$x_{58} = -59.1666616426078$$
$$x_{59} = 47.6474885794452$$
$$x_{60} = -27.7507351067098$$
$$x_{61} = 75.9218224617533$$
$$x_{62} = 41.3643032722656$$
$$x_{63} = -21.4675497995303$$
$$x_{64} = -100.007366139275$$
$$x_{65} = 50.789081233035$$
$$x_{66} = 63.3554518473942$$
The values of the extrema at the points:
/ pi\
(9.94837673636768, -sin|9.94837673636768 + --|)
\ 3 /
/ pi\
(-56.02506898901798, -sin|56.025068989018 - --|)
\ 3 /
/ pi\
(57.07226654021458, sin|57.0722665402146 + --|)
\ 3 /
/ pi\
(-15.184364492350667, sin|15.1843644923507 - --|)
\ 3 /
/ pi\
(60.21385919380437, -sin|60.2138591938044 + --|)
\ 3 /
/ pi\
(82.20500776893293, sin|82.2050077689329 + --|)
\ 3 /
/ pi\
(-37.17551306747922, -sin|37.1755130674792 - --|)
\ 3 /
/ pi\
(-52.883476335428185, sin|52.8834763354282 - --|)
\ 3 /
/ pi\
(85.34660042252271, -sin|85.3466004225227 + --|)
\ 3 /
/ pi\
(-84.29940287132612, sin|84.2994028713261 - --|)
\ 3 /
/ pi\
(-87.4409955249159, -sin|87.4409955249159 - --|)
\ 3 /
/ pi\
(38.22271061867582, sin|38.2227106186758 + --|)
\ 3 /
/ pi\
(-90.5825881785057, sin|90.5825881785057 - --|)
\ 3 /
/ pi\
(66.49704450098396, -sin|66.497044500984 + --|)
\ 3 /
/ pi\
(31.939525311496233, sin|31.9395253114962 + --|)
\ 3 /
/ pi\
(16.231562043547264, -sin|16.2315620435473 + --|)
\ 3 /
/ pi\
(22.51474735072685, -sin|22.5147473507269 + --|)
\ 3 /
/ pi\
(28.797932657906436, -sin|28.7979326579064 + --|)
\ 3 /
/ pi\
(-24.609142453120047, -sin|24.60914245312 - --|)
\ 3 /
/ pi\
(0.5235987755982989, sin|0.523598775598299 + --|)
\ 3 /
/ pi\
(-46.6002910282486, sin|46.6002910282486 - --|)
\ 3 /
/ pi\
(-78.01621756414653, sin|78.0162175641465 - --|)
\ 3 /
/ pi\
(88.48819307611251, sin|88.4881930761125 + --|)
\ 3 /
/ pi\
(-68.59143960337715, -sin|68.5914396033772 - --|)
\ 3 /
/ pi\
(-8.901179185171081, sin|8.90117918517108 - --|)
\ 3 /
/ pi\
(13.089969389957473, sin|13.0899693899575 + --|)
\ 3 /
/ pi\
(44.505895925855405, sin|44.5058959258554 + --|)
\ 3 /
/ pi\
(79.06341511534313, -sin|79.0634151153431 + --|)
\ 3 /
/ pi\
(-65.44984694978736, sin|65.4498469497874 - --|)
\ 3 /
/ pi\
(69.63863715457374, sin|69.6386371545737 + --|)
\ 3 /
/ pi\
(-442.44096538056255, sin|442.440965380563 - --|)
\ 3 /
/ pi\
(19.373154697137057, sin|19.3731546971371 + --|)
\ 3 /
/ pi\
(-34.033920413889426, sin|34.0339204138894 - --|)
\ 3 /
/ pi\
(-74.87462491055673, -sin|74.8746249105567 - --|)
\ 3 /
/ pi\
(91.6297857297023, -sin|91.6297857297023 + --|)
\ 3 /
/ pi\
(-49.741883681838395, -sin|49.7418836818384 - --|)
\ 3 /
/ pi\
(-2.6179938779914944, sin|2.61799387799149 - --|)
\ 3 /
/ pi\
(-81.15781021773633, -sin|81.1578102177363 - --|)
\ 3 /
/ pi\
(6.806784082777885, sin|6.80678408277789 + --|)
\ 3 /
/ pi\
(72.78022980816354, -sin|72.7802298081635 + --|)
\ 3 /
/ pi\
(-43.45869837465881, -sin|43.4586983746588 - --|)
\ 3 /
/ pi\
(-5.759586531581288, -sin|5.75958653158129 - --|)
\ 3 /
/ pi\
(-30.892327760299633, -sin|30.8923277602996 - --|)
\ 3 /
/ pi\
(35.08111796508602, -sin|35.081117965086 + --|)
\ 3 /
/ pi\
(-40.31710572106901, sin|40.317105721069 - --|)
\ 3 /
/ pi\
(97.91297103688188, -sin|97.9129710368819 + --|)
\ 3 /
/ pi\
(-96.8657734856853, sin|96.8657734856853 - --|)
\ 3 /
/ pi\
(-12.042771838760874, -sin|12.0427718387609 - --|)
\ 3 /
/ pi\
(-62.30825429619757, -sin|62.3082542961976 - --|)
\ 3 /
/ pi\
(-93.7241808320955, -sin|93.7241808320955 - --|)
\ 3 /
/ pi\
(-18.32595714594046, -sin|18.3259571459405 - --|)
\ 3 /
/ pi\
(101.05456369047168, sin|101.054563690472 + --|)
\ 3 /
/ pi\
(94.7713783832921, sin|94.7713783832921 + --|)
\ 3 /
/ pi\
(53.93067388662478, -sin|53.9306738866248 + --|)
\ 3 /
/ pi\
(25.656340004316643, sin|25.6563400043166 + --|)
\ 3 /
/ pi\
(3.6651914291880923, -sin|3.66519142918809 + --|)
\ 3 /
/ pi\
(-71.73303225696695, sin|71.733032256967 - --|)
\ 3 /
/ pi\
(-59.16666164260777, sin|59.1666616426078 - --|)
\ 3 /
/ pi\
(47.647488579445195, -sin|47.6474885794452 + --|)
\ 3 /
/ pi\
(-27.75073510670984, sin|27.7507351067098 - --|)
\ 3 /
/ pi\
(75.92182246175334, sin|75.9218224617533 + --|)
\ 3 /
/ pi\
(41.36430327226561, -sin|41.3643032722656 + --|)
\ 3 /
/ pi\
(-21.467549799530254, sin|21.4675497995303 - --|)
\ 3 /
/ pi\
(-100.00736613927508, -sin|100.007366139275 - --|)
\ 3 /
/ pi\
(50.78908123303499, sin|50.789081233035 + --|)
\ 3 /
/ pi\
(63.355451847394164, sin|63.3554518473942 + --|)
\ 3 /
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:
The function has no minima
Maxima of the function at points:
$$x_{66} = 9.94837673636768$$
$$x_{66} = -56.025068989018$$
$$x_{66} = 57.0722665402146$$
$$x_{66} = -15.1843644923507$$
$$x_{66} = 60.2138591938044$$
$$x_{66} = 82.2050077689329$$
$$x_{66} = -37.1755130674792$$
$$x_{66} = -52.8834763354282$$
$$x_{66} = 85.3466004225227$$
$$x_{66} = -84.2994028713261$$
$$x_{66} = -87.4409955249159$$
$$x_{66} = 38.2227106186758$$
$$x_{66} = -90.5825881785057$$
$$x_{66} = 66.497044500984$$
$$x_{66} = 31.9395253114962$$
$$x_{66} = 16.2315620435473$$
$$x_{66} = 22.5147473507269$$
$$x_{66} = 28.7979326579064$$
$$x_{66} = -24.60914245312$$
$$x_{66} = 0.523598775598299$$
$$x_{66} = -46.6002910282486$$
$$x_{66} = -78.0162175641465$$
$$x_{66} = 88.4881930761125$$
$$x_{66} = -68.5914396033772$$
$$x_{66} = -8.90117918517108$$
$$x_{66} = 13.0899693899575$$
$$x_{66} = 44.5058959258554$$
$$x_{66} = 79.0634151153431$$
$$x_{66} = -65.4498469497874$$
$$x_{66} = 69.6386371545737$$
$$x_{66} = -442.440965380563$$
$$x_{66} = 19.3731546971371$$
$$x_{66} = -34.0339204138894$$
$$x_{66} = -74.8746249105567$$
$$x_{66} = 91.6297857297023$$
$$x_{66} = -49.7418836818384$$
$$x_{66} = -2.61799387799149$$
$$x_{66} = -81.1578102177363$$
$$x_{66} = 6.80678408277789$$
$$x_{66} = 72.7802298081635$$
$$x_{66} = -43.4586983746588$$
$$x_{66} = -5.75958653158129$$
$$x_{66} = -30.8923277602996$$
$$x_{66} = 35.081117965086$$
$$x_{66} = -40.317105721069$$
$$x_{66} = 97.9129710368819$$
$$x_{66} = -96.8657734856853$$
$$x_{66} = -12.0427718387609$$
$$x_{66} = -62.3082542961976$$
$$x_{66} = -93.7241808320955$$
$$x_{66} = -18.3259571459405$$
$$x_{66} = 101.054563690472$$
$$x_{66} = 94.7713783832921$$
$$x_{66} = 53.9306738866248$$
$$x_{66} = 25.6563400043166$$
$$x_{66} = 3.66519142918809$$
$$x_{66} = -71.733032256967$$
$$x_{66} = -59.1666616426078$$
$$x_{66} = 47.6474885794452$$
$$x_{66} = -27.7507351067098$$
$$x_{66} = 75.9218224617533$$
$$x_{66} = 41.3643032722656$$
$$x_{66} = -21.4675497995303$$
$$x_{66} = -100.007366139275$$
$$x_{66} = 50.789081233035$$
$$x_{66} = 63.3554518473942$$
Decreasing at intervals
$$\left(-\infty, -442.440965380563\right]$$
Increasing at intervals
$$\left[101.054563690472, \infty\right)$$