Mister Exam

Other calculators

Plotting a function graph/ x+sqrtxcosx

Graphing y = x+sqrtxcosx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
             ___       
f(x) = x + \/ x *cos(x)
f(x)=xcos⁡(x)+xf{\left(x \right)} = \sqrt{x} \cos{\left(x \right)} + x 
f = sqrt(x)*cos(x) + x
The graph of the function
02468-8-6-4-2-1010010
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)+x=0\sqrt{x} \cos{\left(x \right)} + x = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x + 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)+1+cos⁡(x)2x=0- \sqrt{x} \sin{\left(x \right)} + 1 + \frac{\cos{\left(x \right)}}{2 \sqrt{x}} = 0
Solve this equation
The roots of this equation
x1=84.7200461068506x_{1} = 84.7200461068506
x2=21.7982825070044x_{2} = 21.7982825070044
x3=46.9881328578849x_{3} = 46.9881328578849
x4=44.1446958001951x_{4} = 44.1446958001951
x5=94.3562071406871x_{5} = 94.3562071406871
x6=78.4330374018336x_{6} = 78.4330374018336
x7=97.2929565639277x_{7} = 97.2929565639277
x8=34.4007372254998x_{8} = 34.4007372254998
x9=72.1455583293193x_{9} = 72.1455583293193
x10=88.0770257646166x_{10} = 88.0770257646166
x11=28.1023797507614x_{11} = 28.1023797507614
x12=37.8755093111662x_{12} = 37.8755093111662
x13=15.4834271109676x_{13} = 15.4834271109676
x14=56.6906897945907x_{14} = 56.6906897945907
x15=69.2427218976821x_{15} = 69.2427218976821
x16=2.68206122525946x_{16} = 2.68206122525946
x17=40.6955962399499x_{17} = 40.6955962399499
x18=53.2790328573571x_{18} = 53.2790328573571
x19=9.14286874974213x_{19} = 9.14286874974213
x20=59.5687272757879x_{20} = 59.5687272757879
x21=65.857502723781x_{21} = 65.857502723781
x22=81.7983135531006x_{22} = 81.7983135531006
x23=50.4166979783446x_{23} = 50.4166979783446
x24=12.8872266727419x_{24} = 12.8872266727419
x25=19.1064585315631x_{25} = 19.1064585315631
x26=62.9661476727656x_{26} = 62.9661476727656
x27=75.520168975823x_{27} = 75.520168975823
x28=31.6105338836827x_{28} = 31.6105338836827
x29=25.3523559763918x_{29} = 25.3523559763918
x30=91.0066650197939x_{30} = 91.0066650197939
x31=100.635781498703x_{31} = 100.635781498703
x32=6.75104176504286x_{32} = 6.75104176504286
The values of the extrema at the points:
(84.7200461068506, 75.5644360576617)

(21.79828250700435, 17.2159851852599)

(46.98813285788495, 40.1964135628577)

(44.1446958001951, 50.7014215532442)

(94.35620714068712, 104.012875610689)

(78.4330374018336, 69.6272348959494)

(97.29295656392769, 87.4750483550338)

(34.40073722549983, 28.6074607167868)

(72.14555832931933, 63.7040452908452)

(88.07702576461655, 97.4027076211084)

(28.102379750761397, 22.87939211233)

(37.87550931116622, 43.9343165901878)

(15.483427110967554, 11.6473043335045)

(56.690689794590654, 64.144205164234)

(69.24272189768206, 77.496204408866)

(2.6820612252594644, 1.21425555206754)

(40.69559623994987, 34.383331130769)

(53.27903285735705, 46.0395372966015)

(9.142868749742126, 6.23850907997821)

(59.56872727578785, 51.9075783817589)

(65.85750272378104, 57.7967239323487)

(81.7983135531006, 90.7808235550524)

(50.416697978344615, 57.4361440556216)

(12.887226672741932, 16.2938984057417)

(19.10645853156311, 23.3340998397661)

(62.96614767276555, 70.8298211871434)

(75.52016897582297, 84.1458683922881)

(31.610533883682674, 37.1267296651837)

(25.352355976391834, 30.2665321917206)

(91.0066650197939, 81.5141284232659)

(100.63578149870287, 110.61246371051)

(6.751041765042861, 9.07009990034565)


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=84.7200461068506x_{1} = 84.7200461068506
x2=21.7982825070044x_{2} = 21.7982825070044
x3=46.9881328578849x_{3} = 46.9881328578849
x4=78.4330374018336x_{4} = 78.4330374018336
x5=97.2929565639277x_{5} = 97.2929565639277
x6=34.4007372254998x_{6} = 34.4007372254998
x7=72.1455583293193x_{7} = 72.1455583293193
x8=28.1023797507614x_{8} = 28.1023797507614
x9=15.4834271109676x_{9} = 15.4834271109676
x10=2.68206122525946x_{10} = 2.68206122525946
x11=40.6955962399499x_{11} = 40.6955962399499
x12=53.2790328573571x_{12} = 53.2790328573571
x13=9.14286874974213x_{13} = 9.14286874974213
x14=59.5687272757879x_{14} = 59.5687272757879
x15=65.857502723781x_{15} = 65.857502723781
x16=91.0066650197939x_{16} = 91.0066650197939
Maxima of the function at points:
x16=44.1446958001951x_{16} = 44.1446958001951
x16=94.3562071406871x_{16} = 94.3562071406871
x16=88.0770257646166x_{16} = 88.0770257646166
x16=37.8755093111662x_{16} = 37.8755093111662
x16=56.6906897945907x_{16} = 56.6906897945907
x16=69.2427218976821x_{16} = 69.2427218976821
x16=81.7983135531006x_{16} = 81.7983135531006
x16=50.4166979783446x_{16} = 50.4166979783446
x16=12.8872266727419x_{16} = 12.8872266727419
x16=19.1064585315631x_{16} = 19.1064585315631
x16=62.9661476727656x_{16} = 62.9661476727656
x16=75.520168975823x_{16} = 75.520168975823
x16=31.6105338836827x_{16} = 31.6105338836827
x16=25.3523559763918x_{16} = 25.3523559763918
x16=100.635781498703x_{16} = 100.635781498703
x16=6.75104176504286x_{16} = 6.75104176504286
Decreasing at intervals
[97.2929565639277,∞)\left[97.2929565639277, \infty\right)
Increasing at intervals
(−∞,2.68206122525946]\left(-\infty, 2.68206122525946\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=80.1230923289863x_{1} = 80.1230923289863
x2=−61.2773734476957x_{2} = -61.2773734476957
x3=−73.8409685283396x_{3} = -73.8409685283396
x4=39.2953468672842x_{4} = 39.2953468672842
x5=−80.1230923289863x_{5} = -80.1230923289863
x6=36.1559611393004x_{6} = 36.1559611393004
x7=−70.6999773315004x_{7} = -70.6999773315004
x8=−83.2642142711524x_{8} = -83.2642142711524
x9=67.559042028453x_{9} = 67.559042028453
x10=−29.8785771570692x_{10} = -29.8785771570692
x11=−17.3363302997334x_{11} = -17.3363302997334
x12=4.91125081295869x_{12} = 4.91125081295869
x13=−2.0090972384408x_{13} = -2.0090972384408
x14=45.5750291575042x_{14} = 45.5750291575042
x15=−14.2073505099925x_{15} = -14.2073505099925
x16=17.3363302997334x_{16} = 17.3363302997334
x17=−48.7152085571549x_{17} = -48.7152085571549
x18=−39.2953468672842x_{18} = -39.2953468672842
x19=−11.0853581860961x_{19} = -11.0853581860961
x20=−124.100967466518x_{20} = -124.100967466518
x21=−76.9820087826371x_{21} = -76.9820087826371
x22=−133.525176756856x_{22} = -133.525176756856
x23=−92.6877714581404x_{23} = -92.6877714581404
x24=−4.91125081295869x_{24} = -4.91125081295869
x25=−23.6042658400483x_{25} = -23.6042658400483
x26=92.6877714581404x_{26} = 92.6877714581404
x27=−67.559042028453x_{27} = -67.559042028453
x28=26.7409029817025x_{28} = 26.7409029817025
x29=−20.4691384083001x_{29} = -20.4691384083001
x30=48.7152085571549x_{30} = 48.7152085571549
x31=7.97819025123437x_{31} = 7.97819025123437
x32=−54.9960510556604x_{32} = -54.9960510556604
x33=11.0853581860961x_{33} = 11.0853581860961
x34=23.6042658400483x_{34} = 23.6042658400483
x35=−89.5465571901753x_{35} = -89.5465571901753
x36=−86.4053704242642x_{36} = -86.4053704242642
x37=89.5465571901753x_{37} = 89.5465571901753
x38=−36.1559611393004x_{38} = -36.1559611393004
x39=20.4691384083001x_{39} = 20.4691384083001
x40=83.2642142711524x_{40} = 83.2642142711524
x41=98.9702720305701x_{41} = 98.9702720305701
x42=29.8785771570692x_{42} = 29.8785771570692
x43=42.4350586138523x_{43} = 42.4350586138523
x44=−33.0169941017832x_{44} = -33.0169941017832
x45=76.9820087826371x_{45} = 76.9820087826371
x46=−98.9702720305701x_{46} = -98.9702720305701
x47=−45.5750291575042x_{47} = -45.5750291575042
x48=−64.4181707871237x_{48} = -64.4181707871237
x49=51.8555589377593x_{49} = 51.8555589377593
x50=−7.97819025123437x_{50} = -7.97819025123437
x51=64.4181707871237x_{51} = 64.4181707871237
x52=33.0169941017832x_{52} = 33.0169941017832
x53=58.136661973445x_{53} = 58.136661973445
x54=73.8409685283396x_{54} = 73.8409685283396
x55=70.6999773315004x_{55} = 70.6999773315004
x56=−26.7409029817025x_{56} = -26.7409029817025
x57=−42.4350586138523x_{57} = -42.4350586138523
x58=2.0090972384408x_{58} = 2.0090972384408
x59=54.9960510556604x_{59} = 54.9960510556604
x60=−58.136661973445x_{60} = -58.136661973445
x61=86.4053704242642x_{61} = 86.4053704242642
x62=−51.8555589377593x_{62} = -51.8555589377593
x63=−95.8290105250036x_{63} = -95.8290105250036
x64=14.2073505099925x_{64} = 14.2073505099925
x65=95.8290105250036x_{65} = 95.8290105250036
x66=61.2773734476957x_{66} = 61.2773734476957

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)+x)=−∞\lim_{x \to -\infty}\left(\sqrt{x} \cos{\left(x \right)} + x\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(xcos⁡(x)+x)=∞\lim_{x \to \infty}\left(\sqrt{x} \cos{\left(x \right)} + x\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x + sqrt(x)*cos(x), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(xcos⁡(x)+xx)y = x \lim_{x \to -\infty}\left(\frac{\sqrt{x} \cos{\left(x \right)} + x}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(xcos⁡(x)+xx)y = x \lim_{x \to \infty}\left(\frac{\sqrt{x} \cos{\left(x \right)} + x}{x}\right)
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)+x=−x+−xcos⁡(x)\sqrt{x} \cos{\left(x \right)} + x = - x + \sqrt{- x} \cos{\left(x \right)}
- No
xcos⁡(x)+x=x−−xcos⁡(x)\sqrt{x} \cos{\left(x \right)} + x = x - \sqrt{- x} \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator