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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
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f(x) = \/ x *cos(x)
f(x)=xcos⁡(x)f{\left(x \right)} = \sqrt{x} \cos{\left(x \right)} 
f = sqrt(x)*cos(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:
xcos⁡(x)=0\sqrt{x} \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
x3=3π2x_{3} = \frac{3 \pi}{2}
Numerical solution
x1=7.85398163397448x_{1} = 7.85398163397448
x2=−86.3937979737193x_{2} = -86.3937979737193
x3=58.1194640914112x_{3} = 58.1194640914112
x4=23.5619449019235x_{4} = 23.5619449019235
x5=−67.5442420521806x_{5} = -67.5442420521806
x6=−4.71238898038469x_{6} = -4.71238898038469
x7=−20.4203522483337x_{7} = -20.4203522483337
x8=0x_{8} = 0
x9=83.2522053201295x_{9} = 83.2522053201295
x10=−29.845130209103x_{10} = -29.845130209103
x11=−39.2699081698724x_{11} = -39.2699081698724
x12=−98.9601685880785x_{12} = -98.9601685880785
x13=98.9601685880785x_{13} = 98.9601685880785
x14=86.3937979737193x_{14} = 86.3937979737193
x15=26.7035375555132x_{15} = 26.7035375555132
x16=−48.6946861306418x_{16} = -48.6946861306418
x17=−89.5353906273091x_{17} = -89.5353906273091
x18=−17.2787595947439x_{18} = -17.2787595947439
x19=20.4203522483337x_{19} = 20.4203522483337
x20=48.6946861306418x_{20} = 48.6946861306418
x21=−64.4026493985908x_{21} = -64.4026493985908
x22=67.5442420521806x_{22} = 67.5442420521806
x23=14.1371669411541x_{23} = 14.1371669411541
x24=−26.7035375555132x_{24} = -26.7035375555132
x25=42.4115008234622x_{25} = 42.4115008234622
x26=−70.6858347057703x_{26} = -70.6858347057703
x27=−32.9867228626928x_{27} = -32.9867228626928
x28=4.71238898038469x_{28} = 4.71238898038469
x29=39.2699081698724x_{29} = 39.2699081698724
x30=73.8274273593601x_{30} = 73.8274273593601
x31=89.5353906273091x_{31} = 89.5353906273091
x32=45.553093477052x_{32} = 45.553093477052
x33=70.6858347057703x_{33} = 70.6858347057703
x34=−95.8185759344887x_{34} = -95.8185759344887
x35=−7.85398163397448x_{35} = -7.85398163397448
x36=76.9690200129499x_{36} = 76.9690200129499
x37=32.9867228626928x_{37} = 32.9867228626928
x38=−23.5619449019235x_{38} = -23.5619449019235
x39=64.4026493985908x_{39} = 64.4026493985908
x40=−36.1283155162826x_{40} = -36.1283155162826
x41=−83.2522053201295x_{41} = -83.2522053201295
x42=−1.5707963267949x_{42} = -1.5707963267949
x43=−58.1194640914112x_{43} = -58.1194640914112
x44=−10.9955742875643x_{44} = -10.9955742875643
x45=1.5707963267949x_{45} = 1.5707963267949
x46=29.845130209103x_{46} = 29.845130209103
x47=−73.8274273593601x_{47} = -73.8274273593601
x48=−92.6769832808989x_{48} = -92.6769832808989
x49=−54.9778714378214x_{49} = -54.9778714378214
x50=80.1106126665397x_{50} = 80.1106126665397
x51=54.9778714378214x_{51} = 54.9778714378214
x52=−76.9690200129499x_{52} = -76.9690200129499
x53=36.1283155162826x_{53} = 36.1283155162826
x54=61.261056745001x_{54} = 61.261056745001
x55=92.6769832808989x_{55} = 92.6769832808989
x56=−61.261056745001x_{56} = -61.261056745001
x57=17.2787595947439x_{57} = 17.2787595947439
x58=10.9955742875643x_{58} = 10.9955742875643
x59=−51.8362787842316x_{59} = -51.8362787842316
x60=−45.553093477052x_{60} = -45.553093477052
x61=−42.4115008234622x_{61} = -42.4115008234622
x62=−80.1106126665397x_{62} = -80.1106126665397
x63=51.8362787842316x_{63} = 51.8362787842316
x64=95.8185759344887x_{64} = 95.8185759344887
x65=−14.1371669411541x_{65} = -14.1371669411541
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(x)*cos(x).
0cos⁡(0)\sqrt{0} \cos{\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
−xsin⁡(x)+cos⁡(x)2x=0- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}} = 0
Solve this equation
The roots of this equation
x1=−9.4774857054208x_{1} = -9.4774857054208
x2=−43.9936619344429x_{2} = -43.9936619344429
x3=62.8398096434599x_{3} = 62.8398096434599
x4=72.26355003974x_{4} = 72.26355003974
x5=−69.1222718113619x_{5} = -69.1222718113619
x6=47.1344973476771x_{6} = 47.1344973476771
x7=−53.4164352526291x_{7} = -53.4164352526291
x8=22.013857636623x_{8} = 22.013857636623
x9=−18.8760383379859x_{9} = -18.8760383379859
x10=−65.9810235167388x_{10} = -65.9810235167388
x11=37.7123693157661x_{11} = 37.7123693157661
x12=9.4774857054208x_{12} = 9.4774857054208
x13=−75.4048544617952x_{13} = -75.4048544617952
x14=−97.3945059759883x_{14} = -97.3945059759883
x15=−40.8529429059734x_{15} = -40.8529429059734
x16=3.29231002128209x_{16} = 3.29231002128209
x17=−50.2754273458806x_{17} = -50.2754273458806
x18=15.7397193560049x_{18} = 15.7397193560049
x19=−6.36162039206566x_{19} = -6.36162039206566
x20=−37.7123693157661x_{20} = -37.7123693157661
x21=0.653271187094403x_{21} = 0.653271187094403
x22=81.6875298021918x_{22} = 81.6875298021918
x23=50.2754273458806x_{23} = 50.2754273458806
x24=−31.43183263459x_{24} = -31.43183263459
x25=56.5575080935408x_{25} = 56.5575080935408
x26=−94.253084424113x_{26} = -94.253084424113
x27=97.3945059759883x_{27} = 97.3945059759883
x28=53.4164352526291x_{28} = 53.4164352526291
x29=−78.5461819355535x_{29} = -78.5461819355535
x30=−84.8288957966139x_{30} = -84.8288957966139
x31=−22.013857636623x_{31} = -22.013857636623
x32=−59.6986356231676x_{32} = -59.6986356231676
x33=−12.6060134442754x_{33} = -12.6060134442754
x34=12.6060134442754x_{34} = 12.6060134442754
x35=−87.970277977177x_{35} = -87.970277977177
x36=87.970277977177x_{36} = 87.970277977177
x37=25.1526172579356x_{37} = 25.1526172579356
x38=−28.2920048800691x_{38} = -28.2920048800691
x39=18.8760383379859x_{39} = 18.8760383379859
x40=6.36162039206566x_{40} = 6.36162039206566
x41=75.4048544617952x_{41} = 75.4048544617952
x42=65.9810235167388x_{42} = 65.9810235167388
x43=−15.7397193560049x_{43} = -15.7397193560049
x44=−56.5575080935408x_{44} = -56.5575080935408
x45=−62.8398096434599x_{45} = -62.8398096434599
x46=43.9936619344429x_{46} = 43.9936619344429
x47=94.253084424113x_{47} = 94.253084424113
x48=84.8288957966139x_{48} = 84.8288957966139
x49=78.5461819355535x_{49} = 78.5461819355535
x50=40.8529429059734x_{50} = 40.8529429059734
x51=−91.1116746699497x_{51} = -91.1116746699497
x52=−47.1344973476771x_{52} = -47.1344973476771
x53=−34.5719807601687x_{53} = -34.5719807601687
x54=−100.535938219808x_{54} = -100.535938219808
x55=91.1116746699497x_{55} = 91.1116746699497
x56=−3.29231002128209x_{56} = -3.29231002128209
x57=31.43183263459x_{57} = 31.43183263459
x58=34.5719807601687x_{58} = 34.5719807601687
x59=−81.6875298021918x_{59} = -81.6875298021918
x60=−25.1526172579356x_{60} = -25.1526172579356
x61=−72.26355003974x_{61} = -72.26355003974
x62=69.1222718113619x_{62} = 69.1222718113619
x63=59.6986356231676x_{63} = 59.6986356231676
x64=28.2920048800691x_{64} = 28.2920048800691
x65=100.535938219808x_{65} = 100.535938219808
The values of the extrema at the points:
(-9.477485705420795, -3.07427725087097*I)

(-43.993661934442905, 6.63234347961736*I)

(62.839809643459915, 7.92690554538958)

(72.26355003974, -8.50059354672143)

(-69.1222718113619, 8.3137630000695*I)

(47.13449734767706, -6.86507057309731)

(-53.41643525262913, -7.3083346567585*I)

(22.013857636622962, -4.69068300028599)

(-18.876038337985854, 4.34313289225214*I)

(-65.9810235167388, -8.12263718050406*I)

(37.712369315766125, 6.14050009006662)

(9.477485705420795, -3.07427725087097)

(-75.40485446179518, 8.68340596604541*I)

(-97.39450597598831, -9.86873543893722*I)

(-40.85294290597337, -6.39115203326596*I)

(3.2923100212820864, -1.79390283516354)

(-50.27542734588058, 7.0901660932241*I)

(15.73971935600487, -3.96533125786786)

(-6.361620392065665, 2.51447081861791*I)

(-37.712369315766125, 6.14050009006662*I)

(0.6532711870944031, 0.641832750676974)

(81.6875298021918, 9.03794608714833)

(50.27542734588058, 7.0901660932241)

(-31.431832634590037, 5.60570075250289*I)

(56.55750809354077, 7.52017873187663)

(-94.25308442411298, 9.70826617196213*I)

(97.39450597598831, -9.86873543893722)

(53.41643525262913, -7.3083346567585)

(-78.54618193555346, -8.86244882770153*I)

(-84.8288957966139, -9.21010036807552*I)

(-22.013857636622962, -4.69068300028599*I)

(-59.698635623167625, -7.72621823510751*I)

(-12.606013444275414, 3.54770528507369*I)

(12.606013444275414, 3.54770528507369)

(-87.970277977177, 9.3790957026809*I)

(87.970277977177, 9.3790957026809)

(25.152617257935617, 5.01424788582548)

(-28.292004880069126, -5.31819247681142*I)

(18.876038337985854, 4.34313289225214)

(6.361620392065665, 2.51447081861791)

(75.40485446179518, 8.68340596604541)

(65.9810235167388, -8.12263718050406)

(-15.73971935600487, -3.96533125786786*I)

(-56.55750809354077, 7.52017873187663*I)

(-62.839809643459915, 7.92690554538958*I)

(43.993661934442905, 6.63234347961736)

(94.25308442411298, 9.70826617196213)

(84.8288957966139, -9.21010036807552)

(78.54618193555346, -8.86244882770153)

(40.85294290597337, -6.39115203326596)

(-91.11167466994975, -9.54509983536653*I)

(-47.13449734767706, -6.86507057309731*I)

(-34.57198076016866, -5.87917944809784*I)

(-100.53593821980844, 10.0266371036526*I)

(91.11167466994975, -9.54509983536653)

(-3.2923100212820864, -1.79390283516354*I)

(31.431832634590037, 5.60570075250289)

(34.57198076016866, -5.87917944809784)

(-81.6875298021918, 9.03794608714833*I)

(-25.152617257935617, 5.01424788582548*I)

(-72.26355003974, -8.50059354672143*I)

(69.1222718113619, 8.3137630000695)

(59.698635623167625, -7.72621823510751)

(28.292004880069126, -5.31819247681142)

(100.53593821980844, 10.0266371036526)


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=72.26355003974x_{1} = 72.26355003974
x2=47.1344973476771x_{2} = 47.1344973476771
x3=22.013857636623x_{3} = 22.013857636623
x4=9.4774857054208x_{4} = 9.4774857054208
x5=3.29231002128209x_{5} = 3.29231002128209
x6=15.7397193560049x_{6} = 15.7397193560049
x7=97.3945059759883x_{7} = 97.3945059759883
x8=53.4164352526291x_{8} = 53.4164352526291
x9=65.9810235167388x_{9} = 65.9810235167388
x10=84.8288957966139x_{10} = 84.8288957966139
x11=78.5461819355535x_{11} = 78.5461819355535
x12=40.8529429059734x_{12} = 40.8529429059734
x13=91.1116746699497x_{13} = 91.1116746699497
x14=34.5719807601687x_{14} = 34.5719807601687
x15=59.6986356231676x_{15} = 59.6986356231676
x16=28.2920048800691x_{16} = 28.2920048800691
Maxima of the function at points:
x16=62.8398096434599x_{16} = 62.8398096434599
x16=37.7123693157661x_{16} = 37.7123693157661
x16=0.653271187094403x_{16} = 0.653271187094403
x16=81.6875298021918x_{16} = 81.6875298021918
x16=50.2754273458806x_{16} = 50.2754273458806
x16=56.5575080935408x_{16} = 56.5575080935408
x16=12.6060134442754x_{16} = 12.6060134442754
x16=87.970277977177x_{16} = 87.970277977177
x16=25.1526172579356x_{16} = 25.1526172579356
x16=18.8760383379859x_{16} = 18.8760383379859
x16=6.36162039206566x_{16} = 6.36162039206566
x16=75.4048544617952x_{16} = 75.4048544617952
x16=43.9936619344429x_{16} = 43.9936619344429
x16=94.253084424113x_{16} = 94.253084424113
x16=31.43183263459x_{16} = 31.43183263459
x16=69.1222718113619x_{16} = 69.1222718113619
x16=100.535938219808x_{16} = 100.535938219808
Decreasing at intervals
[97.3945059759883,∞)\left[97.3945059759883, \infty\right)
Increasing at intervals
(−∞,3.29231002128209]\left(-\infty, 3.29231002128209\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
−(xcos⁡(x)+sin⁡(x)x+cos⁡(x)4x32)=0- (\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{\sqrt{x}} + \frac{\cos{\left(x \right)}}{4 x^{\frac{3}{2}}}) = 0
Solve this equation
The roots of this equation
x1=−36.1559611393004x_{1} = -36.1559611393004
x2=26.7409029817025x_{2} = 26.7409029817025
x3=80.1230923289863x_{3} = 80.1230923289863
x4=86.4053704242642x_{4} = 86.4053704242642
x5=42.4350586138523x_{5} = 42.4350586138523
x6=−76.9820087826371x_{6} = -76.9820087826371
x7=−7.97819025123437x_{7} = -7.97819025123437
x8=−70.6999773315004x_{8} = -70.6999773315004
x9=−33.0169941017832x_{9} = -33.0169941017832
x10=67.559042028453x_{10} = 67.559042028453
x11=−67.559042028453x_{11} = -67.559042028453
x12=−11.0853581860961x_{12} = -11.0853581860961
x13=51.8555589377593x_{13} = 51.8555589377593
x14=58.136661973445x_{14} = 58.136661973445
x15=33.0169941017832x_{15} = 33.0169941017832
x16=−48.7152085571549x_{16} = -48.7152085571549
x17=−26.7409029817025x_{17} = -26.7409029817025
x18=7.97819025123437x_{18} = 7.97819025123437
x19=−14.2073505099925x_{19} = -14.2073505099925
x20=−73.8409685283396x_{20} = -73.8409685283396
x21=−23.6042658400483x_{21} = -23.6042658400483
x22=4.91125081295869x_{22} = 4.91125081295869
x23=89.5465571901753x_{23} = 89.5465571901753
x24=36.1559611393004x_{24} = 36.1559611393004
x25=64.4181707871237x_{25} = 64.4181707871237
x26=−54.9960510556604x_{26} = -54.9960510556604
x27=76.9820087826371x_{27} = 76.9820087826371
x28=2.0090972384408x_{28} = 2.0090972384408
x29=17.3363302997334x_{29} = 17.3363302997334
x30=−89.5465571901753x_{30} = -89.5465571901753
x31=11.0853581860961x_{31} = 11.0853581860961
x32=−124.100967466518x_{32} = -124.100967466518
x33=−80.1230923289863x_{33} = -80.1230923289863
x34=−45.5750291575042x_{34} = -45.5750291575042
x35=−64.4181707871237x_{35} = -64.4181707871237
x36=54.9960510556604x_{36} = 54.9960510556604
x37=−92.6877714581404x_{37} = -92.6877714581404
x38=92.6877714581404x_{38} = 92.6877714581404
x39=29.8785771570692x_{39} = 29.8785771570692
x40=20.4691384083001x_{40} = 20.4691384083001
x41=14.2073505099925x_{41} = 14.2073505099925
x42=83.2642142711524x_{42} = 83.2642142711524
x43=−58.136661973445x_{43} = -58.136661973445
x44=70.6999773315004x_{44} = 70.6999773315004
x45=−51.8555589377593x_{45} = -51.8555589377593
x46=−83.2642142711524x_{46} = -83.2642142711524
x47=−95.8290105250036x_{47} = -95.8290105250036
x48=−42.4350586138523x_{48} = -42.4350586138523
x49=39.2953468672842x_{49} = 39.2953468672842
x50=23.6042658400483x_{50} = 23.6042658400483
x51=−133.525176756856x_{51} = -133.525176756856
x52=45.5750291575042x_{52} = 45.5750291575042
x53=−61.2773734476957x_{53} = -61.2773734476957
x54=95.8290105250036x_{54} = 95.8290105250036
x55=−39.2953468672842x_{55} = -39.2953468672842
x56=−86.4053704242642x_{56} = -86.4053704242642
x57=−2.0090972384408x_{57} = -2.0090972384408
x58=−98.9702720305701x_{58} = -98.9702720305701
x59=−4.91125081295869x_{59} = -4.91125081295869
x60=−20.4691384083001x_{60} = -20.4691384083001
x61=61.2773734476957x_{61} = 61.2773734476957
x62=98.9702720305701x_{62} = 98.9702720305701
x63=−29.8785771570692x_{63} = -29.8785771570692
x64=−17.3363302997334x_{64} = -17.3363302997334
x65=73.8409685283396x_{65} = 73.8409685283396
x66=48.7152085571549x_{66} = 48.7152085571549

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
[95.8290105250036,∞)\left[95.8290105250036, \infty\right)
Convex at the intervals
(−∞,2.0090972384408]\left(-\infty, 2.0090972384408\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xcos⁡(x))=⟨−∞,∞⟩i\lim_{x \to -\infty}\left(\sqrt{x} \cos{\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→∞(xcos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\sqrt{x} \cos{\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)*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{\sqrt{x}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\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:
xcos⁡(x)=−xcos⁡(x)\sqrt{x} \cos{\left(x \right)} = \sqrt{- x} \cos{\left(x \right)}
- No
xcos⁡(x)=−−xcos⁡(x)\sqrt{x} \cos{\left(x \right)} = - \sqrt{- x} \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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