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Plotting a function graph/ sqrt(x)*sin(x)

Graphing y = sqrt(x)*sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
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f(x) = \/ x *sin(x)
f(x)=xsin⁡(x)f{\left(x \right)} = \sqrt{x} \sin{\left(x \right)} 
f = sqrt(x)*sin(x)
The graph of the function
02468-8-6-4-2-10105-5
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:
xsin⁡(x)=0\sqrt{x} \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=62.8318530717959x_{1} = 62.8318530717959
x2=141.371669411541x_{2} = 141.371669411541
x3=−50.2654824574367x_{3} = -50.2654824574367
x4=47.1238898038469x_{4} = 47.1238898038469
x5=84.8230016469244x_{5} = 84.8230016469244
x6=−53.4070751110265x_{6} = -53.4070751110265
x7=−59.6902604182061x_{7} = -59.6902604182061
x8=91.106186954104x_{8} = 91.106186954104
x9=−84.8230016469244x_{9} = -84.8230016469244
x10=25.1327412287183x_{10} = 25.1327412287183
x11=−3.14159265358979x_{11} = -3.14159265358979
x12=−6.28318530717959x_{12} = -6.28318530717959
x13=−40.8407044966673x_{13} = -40.8407044966673
x14=−18.8495559215388x_{14} = -18.8495559215388
x15=78.5398163397448x_{15} = 78.5398163397448
x16=−75.398223686155x_{16} = -75.398223686155
x17=−9.42477796076938x_{17} = -9.42477796076938
x18=72.2566310325652x_{18} = 72.2566310325652
x19=−43.9822971502571x_{19} = -43.9822971502571
x20=31.4159265358979x_{20} = 31.4159265358979
x21=9.42477796076938x_{21} = 9.42477796076938
x22=40.8407044966673x_{22} = 40.8407044966673
x23=−69.1150383789755x_{23} = -69.1150383789755
x24=12.5663706143592x_{24} = 12.5663706143592
x25=87.9645943005142x_{25} = 87.9645943005142
x26=59.6902604182061x_{26} = 59.6902604182061
x27=−37.6991118430775x_{27} = -37.6991118430775
x28=−100.530964914873x_{28} = -100.530964914873
x29=−91.106186954104x_{29} = -91.106186954104
x30=97.3893722612836x_{30} = 97.3893722612836
x31=0x_{31} = 0
x32=−12.5663706143592x_{32} = -12.5663706143592
x33=−78.5398163397448x_{33} = -78.5398163397448
x34=18.8495559215388x_{34} = 18.8495559215388
x35=34.5575191894877x_{35} = 34.5575191894877
x36=−94.2477796076938x_{36} = -94.2477796076938
x37=43.9822971502571x_{37} = 43.9822971502571
x38=−31.4159265358979x_{38} = -31.4159265358979
x39=−81.6814089933346x_{39} = -81.6814089933346
x40=−65.9734457253857x_{40} = -65.9734457253857
x41=75.398223686155x_{41} = 75.398223686155
x42=56.5486677646163x_{42} = 56.5486677646163
x43=−223.053078404875x_{43} = -223.053078404875
x44=3.14159265358979x_{44} = 3.14159265358979
x45=15.707963267949x_{45} = 15.707963267949
x46=−56.5486677646163x_{46} = -56.5486677646163
x47=−21.9911485751286x_{47} = -21.9911485751286
x48=50.2654824574367x_{48} = 50.2654824574367
x49=−15.707963267949x_{49} = -15.707963267949
x50=28.2743338823081x_{50} = 28.2743338823081
x51=94.2477796076938x_{51} = 94.2477796076938
x52=−62.8318530717959x_{52} = -62.8318530717959
x53=69.1150383789755x_{53} = 69.1150383789755
x54=−34.5575191894877x_{54} = -34.5575191894877
x55=−97.3893722612836x_{55} = -97.3893722612836
x56=21.9911485751286x_{56} = 21.9911485751286
x57=65.9734457253857x_{57} = 65.9734457253857
x58=37.6991118430775x_{58} = 37.6991118430775
x59=−87.9645943005142x_{59} = -87.9645943005142
x60=−72.2566310325652x_{60} = -72.2566310325652
x61=−25.1327412287183x_{61} = -25.1327412287183
x62=−28.2743338823081x_{62} = -28.2743338823081
x63=81.6814089933346x_{63} = 81.6814089933346
x64=6.28318530717959x_{64} = 6.28318530717959
x65=100.530964914873x_{65} = 100.530964914873
x66=53.4070751110265x_{66} = 53.4070751110265
x67=−47.1238898038469x_{67} = -47.1238898038469
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(x)*sin(x).
0sin⁡(0)\sqrt{0} \sin{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
xcos⁡(x)+sin⁡(x)2x=0\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}} = 0
Solve this equation
The roots of this equation
x1=−86.3995849739529x_{1} = -86.3995849739529
x2=17.3076405374146x_{2} = 17.3076405374146
x3=−1.83659720315213x_{3} = -1.83659720315213
x4=11.0408298179713x_{4} = 11.0408298179713
x5=51.8459224452234x_{5} = 51.8459224452234
x6=−92.682377997352x_{6} = -92.682377997352
x7=−76.9755154935569x_{7} = -76.9755154935569
x8=26.7222463741877x_{8} = 26.7222463741877
x9=−95.8237937978449x_{9} = -95.8237937978449
x10=−36.1421488970061x_{10} = -36.1421488970061
x11=86.3995849739529x_{11} = 86.3995849739529
x12=67.5516436614121x_{12} = 67.5516436614121
x13=−58.1280655761511x_{13} = -58.1280655761511
x14=−51.8459224452234x_{14} = -51.8459224452234
x15=−61.2692172687226x_{15} = -61.2692172687226
x16=−26.7222463741877x_{16} = -26.7222463741877
x17=39.2826357527234x_{17} = 39.2826357527234
x18=45.5640665961997x_{18} = 45.5640665961997
x19=98.9652208250325x_{19} = 98.9652208250325
x20=7.91705268466621x_{20} = 7.91705268466621
x21=−29.861872403816x_{21} = -29.861872403816
x22=29.861872403816x_{22} = 29.861872403816
x23=−48.7049516666752x_{23} = -48.7049516666752
x24=−89.5409746049841x_{24} = -89.5409746049841
x25=61.2692172687226x_{25} = 61.2692172687226
x26=1.83659720315213x_{26} = 1.83659720315213
x27=−54.9869642514883x_{27} = -54.9869642514883
x28=−42.4232862577008x_{28} = -42.4232862577008
x29=−64.410411962776x_{29} = -64.410411962776
x30=−11.0408298179713x_{30} = -11.0408298179713
x31=83.2582106616487x_{31} = 83.2582106616487
x32=76.9755154935569x_{32} = 76.9755154935569
x33=20.4448034666183x_{33} = 20.4448034666183
x34=−73.8341991854591x_{34} = -73.8341991854591
x35=−39.2826357527234x_{35} = -39.2826357527234
x36=−67.5516436614121x_{36} = -67.5516436614121
x37=−80.1168534696549x_{37} = -80.1168534696549
x38=33.0018723591446x_{38} = 33.0018723591446
x39=54.9869642514883x_{39} = 54.9869642514883
x40=−14.1724320747999x_{40} = -14.1724320747999
x41=73.8341991854591x_{41} = 73.8341991854591
x42=−4.81584231784594x_{42} = -4.81584231784594
x43=−7.91705268466621x_{43} = -7.91705268466621
x44=64.410411962776x_{44} = 64.410411962776
x45=−83.2582106616487x_{45} = -83.2582106616487
x46=−98.9652208250325x_{46} = -98.9652208250325
x47=36.1421488970061x_{47} = 36.1421488970061
x48=48.7049516666752x_{48} = 48.7049516666752
x49=−20.4448034666183x_{49} = -20.4448034666183
x50=14.1724320747999x_{50} = 14.1724320747999
x51=58.1280655761511x_{51} = 58.1280655761511
x52=−23.5831433102848x_{52} = -23.5831433102848
x53=4.81584231784594x_{53} = 4.81584231784594
x54=80.1168534696549x_{54} = 80.1168534696549
x55=−70.692907433161x_{55} = -70.692907433161
x56=23.5831433102848x_{56} = 23.5831433102848
x57=−45.5640665961997x_{57} = -45.5640665961997
x58=42.4232862577008x_{58} = 42.4232862577008
x59=−17.3076405374146x_{59} = -17.3076405374146
x60=95.8237937978449x_{60} = 95.8237937978449
x61=92.682377997352x_{61} = 92.682377997352
x62=70.692907433161x_{62} = 70.692907433161
x63=−33.0018723591446x_{63} = -33.0018723591446
x64=89.5409746049841x_{64} = 89.5409746049841
The values of the extrema at the points:
(-86.3995849739529, 9.29498206229774*I)

(17.307640537414635, -4.15851032158028)

(-1.8365972031521258, -1.30761941299144*I)

(11.040829817971295, -3.31937237072132)

(51.84592244522343, 7.20007645193272)

(-92.68237799735202, 9.6270286533*I)

(-76.97551549355693, -8.77338405887965*I)

(26.72224637418772, 5.16845181340769)

(-95.82379379784489, -9.78882959875799*I)

(-36.142148897006074, 6.01125886058877*I)

(86.3995849739529, -9.29498206229774)

(67.5516436614121, -8.21875556224649)

(-58.12806557615112, -7.6238943490782*I)

(-51.84592244522343, -7.20007645193272*I)

(-61.269217268722585, 7.82720494097395*I)

(-26.72224637418772, -5.16845181340769*I)

(39.282635752723394, 6.26707847792961)

(45.56406659619972, 6.74970965872142)

(98.96522082503246, -9.94799953505937)

(7.917052684666207, 2.808131180007)

(-29.861872403816044, 5.46383591176171*I)

(29.861872403816044, -5.46383591176171)

(-48.70495166667517, 6.97852557917854*I)

(-89.54097460498406, -9.46246176606193*I)

(61.269217268722585, -7.82720494097395)

(1.8365972031521258, 1.30761941299144)

(-54.98696425148828, 7.4150130205716*I)

(-42.423286257700816, 6.51286373926386*I)

(-64.41041196277601, -8.02536795646149*I)

(-11.040829817971295, 3.31937237072132*I)

(83.25821066164869, 9.12442919108264)

(76.97551549355693, 8.77338405887965)

(20.4448034666183, 4.52024144595309)

(-73.83419918545908, 8.59248586707723*I)

(-39.282635752723394, -6.26707847792961*I)

(-67.5516436614121, 8.21875556224649*I)

(-80.11685346965491, 8.95062752823053*I)

(33.00187235914463, 5.74406639671223)

(54.98696425148828, -7.4150130205716)

(-14.172432074799941, -3.76228841574689*I)

(73.83419918545908, -8.59248586707723)

(-4.815842317845935, 2.18276978467772*I)

(-7.917052684666207, -2.808131180007*I)

(64.41041196277601, 8.02536795646149)

(-83.25821066164869, -9.12442919108264*I)

(-98.96522082503246, 9.94799953505937*I)

(36.142148897006074, -6.01125886058877)

(48.70495166667517, -6.97852557917854)

(-20.4448034666183, -4.52024144595309*I)

(14.172432074799941, 3.76228841574689)

(58.12806557615112, 7.6238943490782)

(-23.583143310284843, 4.85515677204621*I)

(4.815842317845935, -2.18276978467772)

(80.11685346965491, -8.95062752823053)

(-70.692907433161, -8.40769713937167*I)

(23.583143310284843, -4.85515677204621)

(-45.56406659619972, -6.74970965872142*I)

(42.423286257700816, -6.51286373926386)

(-17.307640537414635, 4.15851032158028*I)

(95.82379379784489, 9.78882959875799)

(92.68237799735202, -9.6270286533)

(70.692907433161, 8.40769713937167)

(-33.00187235914463, -5.74406639671223*I)

(89.54097460498406, 9.46246176606193)


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:
x1=17.3076405374146x_{1} = 17.3076405374146
x2=11.0408298179713x_{2} = 11.0408298179713
x3=86.3995849739529x_{3} = 86.3995849739529
x4=67.5516436614121x_{4} = 67.5516436614121
x5=98.9652208250325x_{5} = 98.9652208250325
x6=29.861872403816x_{6} = 29.861872403816
x7=61.2692172687226x_{7} = 61.2692172687226
x8=54.9869642514883x_{8} = 54.9869642514883
x9=73.8341991854591x_{9} = 73.8341991854591
x10=36.1421488970061x_{10} = 36.1421488970061
x11=48.7049516666752x_{11} = 48.7049516666752
x12=4.81584231784594x_{12} = 4.81584231784594
x13=80.1168534696549x_{13} = 80.1168534696549
x14=23.5831433102848x_{14} = 23.5831433102848
x15=42.4232862577008x_{15} = 42.4232862577008
x16=92.682377997352x_{16} = 92.682377997352
Maxima of the function at points:
x16=51.8459224452234x_{16} = 51.8459224452234
x16=26.7222463741877x_{16} = 26.7222463741877
x16=39.2826357527234x_{16} = 39.2826357527234
x16=45.5640665961997x_{16} = 45.5640665961997
x16=7.91705268466621x_{16} = 7.91705268466621
x16=1.83659720315213x_{16} = 1.83659720315213
x16=83.2582106616487x_{16} = 83.2582106616487
x16=76.9755154935569x_{16} = 76.9755154935569
x16=20.4448034666183x_{16} = 20.4448034666183
x16=33.0018723591446x_{16} = 33.0018723591446
x16=64.410411962776x_{16} = 64.410411962776
x16=14.1724320747999x_{16} = 14.1724320747999
x16=58.1280655761511x_{16} = 58.1280655761511
x16=95.8237937978449x_{16} = 95.8237937978449
x16=70.692907433161x_{16} = 70.692907433161
x16=89.5409746049841x_{16} = 89.5409746049841
Decreasing at intervals
[98.9652208250325,∞)\left[98.9652208250325, \infty\right)
Increasing at intervals
(−∞,4.81584231784594]\left(-\infty, 4.81584231784594\right]
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
−xsin⁡(x)+cos⁡(x)x−sin⁡(x)4x32=0- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin{\left(x \right)}}{4 x^{\frac{3}{2}}} = 0
Solve this equation
The roots of this equation
x1=−22.0364734735106x_{1} = -22.0364734735106
x2=−116.247530144815x_{2} = -116.247530144815
x3=91.1171610640786x_{3} = 91.1171610640786
x4=22.0364734735106x_{4} = 22.0364734735106
x5=62.8477621879326x_{5} = 62.8477621879326
x6=−50.2853643733782x_{6} = -50.2853643733782
x7=−65.9885978289116x_{7} = -65.9885978289116
x8=6.43640901362357x_{8} = 6.43640901362357
x9=75.4114829061337x_{9} = 75.4114829061337
x10=−31.4477066173312x_{10} = -31.4477066173312
x11=−9.52905247096223x_{11} = -9.52905247096223
x12=−100.54091054091x_{12} = -100.54091054091
x13=15.7712217163826x_{13} = 15.7712217163826
x14=−91.1171610640786x_{14} = -91.1171610640786
x15=18.9023731724419x_{15} = 18.9023731724419
x16=72.2704663982901x_{16} = 72.2704663982901
x17=34.5864181840427x_{17} = 34.5864181840427
x18=78.5525454686572x_{18} = 78.5525454686572
x19=56.5663428995631x_{19} = 56.5663428995631
x20=−81.6936487772184x_{20} = -81.6936487772184
x21=−56.5663428995631x_{21} = -56.5663428995631
x22=−3.42038548945687x_{22} = -3.42038548945687
x23=−37.7256081789305x_{23} = -37.7256081789305
x24=40.8651666720526x_{24} = 40.8651666720526
x25=31.4477066173312x_{25} = 31.4477066173312
x26=−18.9023731724419x_{26} = -18.9023731724419
x27=−78.5525454686572x_{27} = -78.5525454686572
x28=−6.43640901362357x_{28} = -6.43640901362357
x29=−62.8477621879326x_{29} = -62.8477621879326
x30=25.1724307086655x_{30} = 25.1724307086655
x31=81.6936487772184x_{31} = 81.6936487772184
x32=−94.2583880465909x_{32} = -94.2583880465909
x33=44.0050149904158x_{33} = 44.0050149904158
x34=87.9759601854462x_{34} = 87.9759601854462
x35=−72.2704663982901x_{35} = -72.2704663982901
x36=−87.9759601854462x_{36} = -87.9759601854462
x37=100.54091054091x_{37} = 100.54091054091
x38=50.2853643733782x_{38} = 50.2853643733782
x39=−34.5864181840427x_{39} = -34.5864181840427
x40=−40.8651666720526x_{40} = -40.8651666720526
x41=65.9885978289116x_{41} = 65.9885978289116
x42=−75.4114829061337x_{42} = -75.4114829061337
x43=9.52905247096223x_{43} = 9.52905247096223
x44=0.746349736778129x_{44} = 0.746349736778129
x45=−97.3996386085752x_{45} = -97.3996386085752
x46=53.4257888392775x_{46} = 53.4257888392775
x47=84.834788308704x_{47} = 84.834788308704
x48=−69.1295022175061x_{48} = -69.1295022175061
x49=−28.3096318664276x_{49} = -28.3096318664276
x50=59.7070061315463x_{50} = 59.7070061315463
x51=47.1450953533935x_{51} = 47.1450953533935
x52=28.3096318664276x_{52} = 28.3096318664276
x53=3.42038548945687x_{53} = 3.42038548945687
x54=−84.834788308704x_{54} = -84.834788308704
x55=97.3996386085752x_{55} = 97.3996386085752
x56=−25.1724307086655x_{56} = -25.1724307086655
x57=−53.4257888392775x_{57} = -53.4257888392775
x58=37.7256081789305x_{58} = 37.7256081789305
x59=−15.7712217163826x_{59} = -15.7712217163826
x60=12.64516529855x_{60} = 12.64516529855
x61=−47.1450953533935x_{61} = -47.1450953533935
x62=−59.7070061315463x_{62} = -59.7070061315463
x63=−0.746349736778129x_{63} = -0.746349736778129
x64=69.1295022175061x_{64} = 69.1295022175061
x65=−44.0050149904158x_{65} = -44.0050149904158
x66=94.2583880465909x_{66} = 94.2583880465909
x67=−12.64516529855x_{67} = -12.64516529855

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
[97.3996386085752,∞)\left[97.3996386085752, \infty\right)
Convex at the intervals
(−∞,3.42038548945687]\left(-\infty, 3.42038548945687\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xsin⁡(x))=⟨−∞,∞⟩i\lim_{x \to -\infty}\left(\sqrt{x} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle i
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩iy = \left\langle -\infty, \infty\right\rangle i
lim⁡x→∞(xsin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\sqrt{x} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(x)*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{x}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{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:
xsin⁡(x)=−−xsin⁡(x)\sqrt{x} \sin{\left(x \right)} = - \sqrt{- x} \sin{\left(x \right)}
- No
xsin⁡(x)=−xsin⁡(x)\sqrt{x} \sin{\left(x \right)} = \sqrt{- x} \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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