Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • x/x^2+1 x/x^2+1
  • x/(x^2-1)
  • xe^(1/x)
  • xcosx

  • Identical expressions

  • exp(six *x-x^ two)
  • exponent of (6 multiply by x minus x squared )
  • exponent of (six multiply by x minus x to the power of two)
  • exp(6*x-x2)
  • exp6*x-x2
  • exp(6*x-x²)
  • exp(6*x-x to the power of 2)
  • exp(6x-x^2)
  • exp(6x-x2)
  • exp6x-x2
  • exp6x-x^2

  • Similar expressions

  • exp(6*x+x^2)
Plotting a function graph/ exp(6*x-x^2)

Graphing y = exp(6*x-x^2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
               2
        6*x - x 
f(x) = e
f(x)=e−x2+6xf{\left(x \right)} = e^{- x^{2} + 6 x} 
f = exp(-x^2 + 6*x)
The graph of the function
02468-8-6-4-2-1010010000
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:
e−x2+6x=0e^{- x^{2} + 6 x} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to exp(6*x - x^2).
e0⋅6−02e^{0 \cdot 6 - 0^{2}}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
(6−2x)e−x2+6x=0\left(6 - 2 x\right) e^{- x^{2} + 6 x} = 0
Solve this equation
The roots of this equation
x1=52.4560161379821x_{1} = 52.4560161379821
x2=34.5738300131898x_{2} = 34.5738300131898
x3=74.392536498399x_{3} = 74.392536498399
x4=24.7238244881469x_{4} = 24.7238244881469
x5=−16.5109382379574x_{5} = -16.5109382379574
x6=−40.2278099160914x_{6} = -40.2278099160914
x7=58.4337104086693x_{7} = 58.4337104086693
x8=30.6208538623602x_{8} = 30.6208538623602
x9=98.3566616903333x_{9} = 98.3566616903333
x10=16.9992809268135x_{10} = 16.9992809268135
x11=70.400998762587x_{11} = 70.400998762587
x12=−3.74301612101296x_{12} = -3.74301612101296
x13=−60.1556753178057x_{13} = -60.1556753178057
x14=−8.86545060149677x_{14} = -8.86545060149677
x15=86.3720184879914x_{15} = 86.3720184879914
x16=−78.1211308110769x_{16} = -78.1211308110769
x17=−68.1381651356283x_{17} = -68.1381651356283
x18=−46.2000176225044x_{18} = -46.2000176225044
x19=11.5562499528413x_{19} = 11.5562499528413
x20=−26.3368875035311x_{20} = -26.3368875035311
x21=−56.1662052407992x_{21} = -56.1662052407992
x22=84.375018132477x_{22} = 84.375018132477
x23=44.495785122687x_{23} = 44.495785122687
x24=82.3781688739501x_{24} = 82.3781688739501
x25=−96.0050507000099x_{25} = -96.0050507000099
x26=−48.1921992657937x_{26} = -48.1921992657937
x27=−44.2084977857701x_{27} = -44.2084977857701
x28=−76.1241933853779x_{28} = -76.1241933853779
x29=76.3886510740335x_{29} = 76.3886510740335
x30=72.3966457620089x_{30} = 72.3966457620089
x31=54.4480034075534x_{31} = 54.4480034075534
x32=92.3638248016984x_{32} = 92.3638248016984
x33=46.4844728781857x_{33} = 46.4844728781857
x34=90.366431024879x_{34} = 90.366431024879
x35=−7.04433982996113x_{35} = -7.04433982996113
x36=−58.1607682345028x_{36} = -58.1607682345028
x37=18.9046192283974x_{37} = 18.9046192283974
x38=−64.1463989860564x_{38} = -64.1463989860564
x39=56.4405897196054x_{39} = 56.4405897196054
x40=−20.4235611844925x_{40} = -20.4235611844925
x41=−28.3153447286813x_{41} = -28.3153447286813
x42=32.5957583046983x_{42} = 32.5957583046983
x43=22.7719971377255x_{43} = 22.7719971377255
x44=48.4741540144396x_{44} = 48.4741540144396
x45=94.361332630532x_{45} = 94.361332630532
x46=28.6498514574821x_{46} = 28.6498514574821
x47=−22.3901348160467x_{47} = -22.3901348160467
x48=−90.1055171244797x_{48} = -90.1055171244797
x49=50.4647034274775x_{49} = 50.4647034274775
x50=−74.1274147285816x_{50} = -74.1274147285816
x51=−82.1154372504332x_{51} = -82.1154372504332
x52=60.4273097814622x_{52} = 60.4273097814622
x53=9.96154504225972x_{53} = 9.96154504225972
x54=−72.1308075062542x_{54} = -72.1308075062542
x55=80.3814824148947x_{55} = 80.3814824148947
x56=62.4213396148336x_{56} = 62.4213396148336
x57=−34.2645449807319x_{57} = -34.2645449807319
x58=−100.004844848378x_{58} = -100.004844848378
x59=−88.1078338924457x_{59} = -88.1078338924457
x60=−86.1102546220613x_{60} = -86.1102546220613
x61=36.554507149414x_{61} = 36.554507149414
x62=68.4056178257571x_{62} = 68.4056178257571
x63=−54.1720223810675x_{63} = -54.1720223810675
x64=88.3691593312176x_{64} = 88.3691593312176
x65=−24.3615719551025x_{65} = -24.3615719551025
x66=−30.296381804162x_{66} = -30.296381804162
x67=−94.1011702664187x_{67} = -94.1011702664187
x68=20.8309297171705x_{68} = 20.8309297171705
x69=64.4157579453486x_{69} = 64.4157579453486
x70=−70.134385766936x_{70} = -70.134385766936
x71=78.3849716980399x_{71} = 78.3849716980399
x72=96.3589472776468x_{72} = 96.3589472776468
x73=−62.1508948593432x_{73} = -62.1508948593432
x74=−80.1182155582791x_{74} = -80.1182155582791
x75=−50.1849682625092x_{75} = -50.1849682625092
x76=−84.1127864691674x_{76} = -84.1127864691674
x77=42.5082410274504x_{77} = 42.5082410274504
x78=−66.1421630415863x_{78} = -66.1421630415863
x79=26.6837303756218x_{79} = 26.6837303756218
x80=−5.3115816730434x_{80} = -5.3115816730434
x81=15.125084358924x_{81} = 15.125084358924
x82=−32.2795629354021x_{82} = -32.2795629354021
x83=−38.2388692669542x_{83} = -38.2388692669542
x84=66.4105280865885x_{84} = 66.4105280865885
x85=13.2997190058202x_{85} = 13.2997190058202
x86=−98.0049465224685x_{86} = -98.0049465224685
x87=−18.4631983204608x_{87} = -18.4631983204608
x88=−12.6430797877549x_{88} = -12.6430797877549
x89=−14.5695229622524x_{89} = -14.5695229622524
x90=100.35446952839x_{90} = 100.35446952839
x91=38.5373529142981x_{91} = 38.5373529142981
x92=40.5220226949901x_{92} = 40.5220226949901
x93=−36.2510541282376x_{93} = -36.2510541282376
x94=−10.7381124092375x_{94} = -10.7381124092375
x95=−92.1032979124459x_{95} = -92.1032979124459
x96=−52.1782609089954x_{96} = -52.1782609089954
x97=−42.2177273205401x_{97} = -42.2177273205401
The values of the extrema at the points:
(52.45601613798206, 4.66496923933846e-1059)

(34.573830013189806, 9.06887558477286e-430)

(74.39253649839898, 2.26774929948687e-2210)

(24.72382448814687, 9.00372511156435e-202)

(-16.51093823795736, 3.82697295547156e-162)

(-40.22780991609136, 2.32845509303339e-808)

(58.433710408669306, 2.32683529737887e-1331)

(30.62085386236018, 3.80510054798218e-328)

(98.35666169033328, 8.23051204652814e-3946)

(16.999280926813473, 6.2470612781032e-82)

(70.40099876258702, 9.00697253959946e-1970)

(-3.743016121012964, 1.45222241008765e-16)

(-60.155675317805745, 4.61956681914035e-1729)

(-8.865450601496772, 5.81835407692524e-58)

(86.37201848799135, 1.49270306663335e-3015)

(-78.12113081107688, 9.41756375263768e-2855)

(-68.13816513562826, 1.26257737197309e-2194)

(-46.200017622504404, 4.33813010122174e-1048)

(11.556249952841327, 1.30078684705939e-28)

(-26.33688750353106, 1.3546132821367e-370)

(-56.166205240799194, 3.98183781624933e-1517)

(84.37501813247697, 1.13975236821269e-2872)

(44.495785122686975, 1.24997836024491e-744)

(82.37816887395014, 2.91936279442467e-2733)

(-96.00505070000989, 8.99727258359308e-4254)

(-48.19219926579367, 6.00633580457691e-1135)

(-44.20849778577015, 1.05102560824248e-964)

(-76.12419338537785, 8.87407127801895e-2716)

(76.38865107403352, 6.99132970556868e-2336)

(72.39664576200886, 2.46756332056428e-2088)

(54.448003407553365, 2.37610346767507e-1146)

(92.36382480169839, 4.77867846119486e-3465)

(46.48447287818566, 5.02901719053532e-818)

(90.36643102487902, 9.66545289971107e-3312)

(-7.044339829961129, 1.23941699039713e-40)

(-58.16076823450285, 2.34167084004498e-1621)

(18.904619228397404, 1.12423978105435e-106)

(-64.14639898605638, 6.78666738382049e-1955)

(56.44058971960544, 4.05978677773672e-1237)

(-20.42356118449253, 4.23879185988845e-235)

(-28.315344728681275, 1.04090837282703e-422)

(32.59575830469832, 3.20807294635825e-377)

(22.771997137725517, 1.34615801831863e-166)

(48.4741540144396, 6.78676845170831e-895)

(94.361332630532, 7.92566951610389e-3622)

(28.64985145748212, 1.51303328427346e-282)

(-22.390134816046736, 8.64624205746604e-277)

(-90.1055171244797, 1.46993140115752e-3761)

(50.4647034274775, 3.07220090158847e-975)

(-74.12741472858161, 2.80509444884618e-2580)

(-82.11543725043317, 4.00398032601546e-3143)

(60.42730978146217, 4.47358075843085e-1429)

(9.961545042259718, 7.26757227587269e-18)

(-72.13080750625416, 2.97448557359762e-2448)

(80.38148241489469, 2.50845187222513e-2597)

(62.42133961483356, 2.88518972251391e-1530)

(-34.26454498073191, 6.71768214028047e-600)

(-100.00484484837843, 1.10927417614261e-4604)

(-88.10783389244567, 9.71924937872794e-3602)

(-86.11025462206133, 2.15581007088375e-3445)

(36.55450714941399, 8.59692993335301e-486)

(68.4056178257571, 1.10287371559859e-1854)

(-54.17202238106753, 2.27129133157162e-1416)

(88.3691593312176, 6.55808820525051e-3162)

(-24.361571955102548, 5.91079352047288e-322)

(-30.296381804161957, 2.68218886710039e-478)

(-94.1011702664187, 1.26915120004191e-4091)

(20.830929717170516, 6.73240863936905e-135)

(64.4157579453486, 6.24201497935865e-1635)

(-70.13438576693599, 1.0580728063907e-2319)

(78.3849716980399, 7.23041414534983e-2465)

(96.3589472776468, 4.40960410378626e-3782)

(-62.150894859343246, 3.05711033809837e-1840)

(-80.1182155582791, 3.35270039977046e-2997)

(-50.18496826250919, 2.78957661548699e-1225)

(-84.11278646916743, 1.60409496633871e-3292)

(42.50824102745043, 1.04210592628642e-674)

(-66.14216304158627, 5.0540389078208e-2073)

(26.683730375621803, 2.0164485357226e-240)

(-5.311581673043402, 8.06398299980979e-27)

(15.125084358924044, 1.14732697654718e-60)

(-32.27956293540211, 2.31783607657345e-537)

(-38.23886926695423, 2.12911187714179e-735)

(66.41052808658847, 4.53010387002048e-1743)

(13.299719005820185, 6.86851138871952e-43)

(-98.0049465224685, 1.72458195798904e-4427)

(-18.463198320460812, 6.96140398784859e-197)

(-12.643079787754933, 4.3073347799597e-103)

(-14.569522962252366, 7.03227355521173e-131)

(100.35446952839004, 5.15388258319175e-4113)

(38.5373529142981, 2.73306455268495e-545)

(40.522022694990106, 2.91406617551793e-608)

(-36.25105412823759, 6.53009672901164e-666)

(-10.738112409237461, 8.74506706754012e-79)

(-92.1032979124459, 7.45746972847133e-3925)

(-52.17826090899544, 4.34602101790554e-1319)

(-42.217727320540064, 8.54155661846103e-885)


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
The function has no maxima
Decreasing at the entire real axis
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(2(x−3)2−1)ex(6−x)=02 \left(2 \left(x - 3\right)^{2} - 1\right) e^{x \left(6 - x\right)} = 0
Solve this equation
The roots of this equation
x1=68.4056360477541x_{1} = 68.4056360477541
x2=46.4845351370641x_{2} = 46.4845351370641
x3=−52.1782902574728x_{3} = -52.1782902574728
x4=76.3886639644834x_{4} = 76.3886639644834
x5=−38.2389398145583x_{5} = -38.2389398145583
x6=−46.2000590688826x_{6} = -46.2000590688826
x7=80.3814934082711x_{7} = 80.3814934082711
x8=90.3664386613481x_{8} = 90.3664386613481
x9=18.9059564908195x_{9} = 18.9059564908195
x10=−28.3155067841679x_{10} = -28.3155067841679
x11=82.3781790571961x_{11} = 82.3781790571961
x12=26.6841225372817x_{12} = 26.6841225372817
x13=−80.1182241228293x_{13} = -80.1182241228293
x14=−70.1343983475769x_{14} = -70.1343983475769
x15=64.4157799638918x_{15} = 64.4157799638918
x16=48.4742084194675x_{16} = 48.4742084194675
x17=−48.1922360437359x_{17} = -48.1922360437359
x18=3.70710678118655x_{18} = 3.70710678118655
x19=−64.1464152490962x_{19} = -64.1464152490962
x20=66.4105480873281x_{20} = 66.4105480873281
x21=58.4337403772313x_{21} = 58.4337403772313
x22=−92.1033035165728x_{22} = -92.1033035165728
x23=17.0012777744454x_{23} = 17.0012777744454
x24=−50.1850010470255x_{24} = -50.1850010470255
x25=98.3566674788426x_{25} = 98.3566674788426
x26=−5.32269391999105x_{26} = -5.32269391999105
x27=84.3750275833859x_{27} = 84.3750275833859
x28=−22.3904410852112x_{28} = -22.3904410852112
x29=−24.3618159522672x_{29} = -24.3618159522672
x30=88.369167514995x_{30} = 88.369167514995
x31=72.3966610120181x_{31} = 72.3966610120181
x32=11.5664526708467x_{32} = 11.5664526708467
x33=28.6501590282445x_{33} = 28.6501590282445
x34=52.4560583881637x_{34} = 52.4560583881637
x35=−42.2177807645343x_{35} = -42.2177807645343
x36=96.3589534992602x_{36} = 96.3589534992602
x37=−88.1078404144289x_{37} = -88.1078404144289
x38=−44.2085447242652x_{38} = -44.2085447242652
x39=−54.1720487570141x_{39} = -54.1720487570141
x40=−72.1308191088924x_{40} = -72.1308191088924
x41=−16.5116236618385x_{41} = -16.5116236618385
x42=−100.004872297801x_{42} = -100.004872297801
x43=−96.0050513309806x_{43} = -96.0050513309806
x44=−84.1127939015959x_{44} = -84.1127939015959
x45=−30.2965164060315x_{45} = -30.2965164060315
x46=15.1282450910777x_{46} = 15.1282450910777
x47=−60.1556948702322x_{47} = -60.1556948702322
x48=−98.0049527616576x_{48} = -98.0049527616576
x49=−90.1055231123979x_{49} = -90.1055231123979
x50=−68.1381788072998x_{50} = -68.1381788072998
x51=−7.0500655359739x_{51} = -7.0500655359739
x52=−34.2646407635773x_{52} = -34.2646407635773
x53=30.6210994527089x_{53} = 30.6210994527089
x54=−86.1102615833838x_{54} = -86.1102615833838
x55=−20.4239527325761x_{55} = -20.4239527325761
x56=−3.76791523640854x_{56} = -3.76791523640854
x57=−10.7401632108074x_{57} = -10.7401632108074
x58=40.5221198429226x_{58} = 40.5221198429226
x59=86.3720272748887x_{59} = 86.3720272748887
x60=34.5739937230882x_{60} = 34.5739937230882
x61=−12.6444405159595x_{61} = -12.6444405159595
x62=60.4273367278067x_{62} = 60.4273367278067
x63=−8.86873987654785x_{63} = -8.86873987654785
x64=−78.121140025594x_{64} = -78.121140025594
x65=56.4406231792732x_{65} = 56.4406231792732
x66=−82.1154452259523x_{66} = -82.1154452259523
x67=−58.1607897680498x_{67} = -58.1607897680498
x68=−36.2511360134471x_{68} = -36.2511360134471
x69=−66.1421779340921x_{69} = -66.1421779340921
x70=54.448040921684x_{70} = 54.448040921684
x71=−62.1509126662648x_{71} = -62.1509126662648
x72=−26.3370849881563x_{72} = -26.3370849881563
x73=−94.1011751470352x_{73} = -94.1011751470352
x74=13.3051175196805x_{74} = 13.3051175196805
x75=−74.127425451889x_{75} = -74.127425451889
x76=−32.2796759394231x_{76} = -32.2796759394231
x77=36.5546433285019x_{77} = 36.5546433285019
x78=38.5374673937235x_{78} = 38.5374673937235
x79=−76.1242033161843x_{79} = -76.1242033161843
x80=2.29289321881345x_{80} = 2.29289321881345
x81=42.5083241676641x_{81} = 42.5083241676641
x82=−18.4637097805595x_{82} = -18.4637097805595
x83=92.3638319253346x_{83} = 92.3638319253346
x84=−14.5704704273981x_{84} = -14.5704704273981
x85=44.4958568200158x_{85} = 44.4958568200158
x86=32.5959574669433x_{86} = 32.5959574669433
x87=50.4647512445571x_{87} = 50.4647512445571
x88=70.401015410542x_{88} = 70.401015410542
x89=78.3849835896893x_{89} = 78.3849835896893
x90=24.7243351041127x_{90} = 24.7243351041127
x91=62.4213639317561x_{91} = 62.4213639317561
x92=100.354475198885x_{92} = 100.354475198885
x93=22.7726787189143x_{93} = 22.7726787189143
x94=−40.2278711235207x_{94} = -40.2278711235207
x95=9.98362634704309x_{95} = 9.98362634704309
x96=74.3925505026189x_{96} = 74.3925505026189
x97=−56.1662290321839x_{97} = -56.1662290321839
x98=94.361339304731x_{98} = 94.361339304731
x99=20.8318671133789x_{99} = 20.8318671133789

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,2.29289321881345]∪[3.70710678118655,∞)\left(-\infty, 2.29289321881345\right] \cup \left[3.70710678118655, \infty\right)
Convex at the intervals
[2.29289321881345,3.70710678118655]\left[2.29289321881345, 3.70710678118655\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞e−x2+6x=0\lim_{x \to -\infty} e^{- x^{2} + 6 x} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞e−x2+6x=0\lim_{x \to \infty} e^{- x^{2} + 6 x} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of exp(6*x - x^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(e−x2+6xx)=0\lim_{x \to -\infty}\left(\frac{e^{- x^{2} + 6 x}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(e−x2+6xx)=0\lim_{x \to \infty}\left(\frac{e^{- x^{2} + 6 x}}{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:
e−x2+6x=e−x2−6xe^{- x^{2} + 6 x} = e^{- x^{2} - 6 x}
- No
e−x2+6x=−e−x2−6xe^{- x^{2} + 6 x} = - e^{- x^{2} - 6 x}
- No
so, the function
not is
neither even, nor odd
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