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  • Graphing y =:
  • -x^2-2x+1
  • (x+1)(x-2)^2
  • 6x-2x^2
  • 9^(1/(x-3))

  • Identical expressions

  • x^ three /(two (x- one))^ seven
  • x cubed divide by (2(x minus 1)) to the power of 7
  • x to the power of three divide by (two (x minus one)) to the power of seven
  • x3/(2(x-1))7
  • x3/2x-17
  • x³/(2(x-1))⁷
  • x to the power of 3/(2(x-1)) to the power of 7
  • x^3/2x-1^7
  • x^3 divide by (2(x-1))^7

  • Similar expressions

  • x^3/(2(x+1))^7
Plotting a function graph/ x^3/(2(x-1))^7

Graphing y = x^3/(2(x-1))^7

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
             3     
            x      
f(x) = ------------
                  7
       (2*(x - 1))
f(x)=x3(2(x−1))7f{\left(x \right)} = \frac{x^{3}}{\left(2 \left(x - 1\right)\right)^{7}} 
f = x^3/(2*(x - 1))^7
The graph of the function
02468-8-6-4-2-1010-2500000025000000
The domain of the function
The points at which the function is not precisely defined:
x1=1x_{1} = 1
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:
x3(2(x−1))7=0\frac{x^{3}}{\left(2 \left(x - 1\right)\right)^{7}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=7025.36447009133x_{1} = 7025.36447009133
x2=4191.24285474106x_{2} = 4191.24285474106
x3=283.369677563071x_{3} = 283.369677563071
x4=709.994841915894x_{4} = 709.994841915894
x5=7243.39881762352x_{5} = 7243.39881762352
x6=−3663.84922887189x_{6} = -3663.84922887189
x7=6153.25174085506x_{7} = 6153.25174085506
x8=−7369.72585191428x_{8} = -7369.72585191428
x9=10296.0363390302x_{9} = 10296.0363390302
x10=−1921.87215448153x_{10} = -1921.87215448153
x11=7679.47347638458x_{11} = 7679.47347638458
x12=−5843.56715254363x_{12} = -5843.56715254363
x13=−6061.57921265247x_{13} = -6061.57921265247
x14=6589.30269751786x_{14} = 6589.30269751786
x15=−629.925124877358x_{15} = -629.925124877358
x16=−4099.70477387237x_{16} = -4099.70477387237
x17=−4535.61761919111x_{17} = -4535.61761919111
x18=10077.9839865567x_{18} = 10077.9839865567
x19=−2356.97418233561x_{19} = -2356.97418233561
x20=−3881.76856639939x_{20} = -3881.76856639939
x21=3973.27812144134x_{21} = 3973.27812144134
x22=−7587.75974926522x_{22} = -7587.75974926522
x23=4845.19186054201x_{23} = 4845.19186054201
x24=−8677.95726171392x_{24} = -8677.95726171392
x25=−8896.00138577496x_{25} = -8896.00138577496
x26=3319.46256086385x_{26} = 3319.46256086385
x27=926.121349761496x_{27} = 926.121349761496
x28=2012.53587125297x_{28} = 2012.53587125297
x29=8987.73496311515x_{29} = 8987.73496311515
x30=−11076.5001832165x_{30} = -11076.5001832165
x31=−5189.5614661706x_{31} = -5189.5614661706
x32=−1487.47648714223x_{32} = -1487.47648714223
x33=5063.18934054745x_{33} = 5063.18934054745
x34=3101.55893594303x_{34} = 3101.55893594303
x35=−10858.446557456x_{35} = -10858.446557456
x36=−10640.3936250905x_{36} = -10640.3936250905
x37=4409.21758828009x_{37} = 4409.21758828009
x38=−1054.88223611128x_{38} = -1054.88223611128
x39=8333.59808591774x_{39} = 8333.59808591774
x40=8115.55504323017x_{40} = 8115.55504323017
x41=0x_{41} = 0
x42=6371.2757324461x_{42} = 6371.2757324461
x43=3537.38567460045x_{43} = 3537.38567460045
x44=6807.33235817114x_{44} = 6807.33235817114
x45=−4317.6551716691x_{45} = -4317.6551716691
x46=−1704.54749579971x_{46} = -1704.54749579971
x47=−4753.59038580646x_{47} = -4753.59038580646
x48=−6933.66511393254x_{48} = -6933.66511393254
x49=−5407.55764107557x_{49} = -5407.55764107557
x50=3755.32504616497x_{50} = 3755.32504616497
x51=1794.9064993029x_{51} = 1794.9064993029
x52=10514.0894196925x_{52} = 10514.0894196925
x53=5935.23104020496x_{53} = 5935.23104020496
x54=−9768.18977743347x_{54} = -9768.18977743347
x55=−2574.66831637685x_{55} = -2574.66831637685
x56=5281.1926174337x_{56} = 5281.1926174337
x57=−2139.36471592755x_{57} = -2139.36471592755
x58=495.131558582189x_{58} = 495.131558582189
x59=9641.88165992331x_{59} = 9641.88165992331
x60=−8241.87325934021x_{60} = -8241.87325934021
x61=8769.68814013035x_{61} = 8769.68814013035
x62=−8023.83361761435x_{62} = -8023.83361761435
x63=−6279.59544394292x_{63} = -6279.59544394292
x64=−3010.23171104589x_{64} = -3010.23171104589
x65=4627.20096439039x_{65} = 4627.20096439039
x66=2883.67894787455x_{66} = 2883.67894787455
x67=9423.8317936207x_{67} = 9423.8317936207
x68=2448.01333523232x_{68} = 2448.01333523232
x69=2230.24472366323x_{69} = 2230.24472366323
x70=−2792.42573881182x_{70} = -2792.42573881182
x71=−10204.2900218538x_{71} = -10204.2900218538
x72=−7805.79572918976x_{72} = -7805.79572918976
x73=9859.93240965804x_{73} = 9859.93240965804
x74=1360.02074794152x_{74} = 1360.02074794152
x75=5499.20102485165x_{75} = 5499.20102485165
x76=−7151.69423304465x_{76} = -7151.69423304465
x77=8551.64248454177x_{77} = 8551.64248454177
x78=1577.3860846559x_{78} = 1577.3860846559
x79=−6497.61541327269x_{79} = -6497.61541327269
x80=−9332.09337289399x_{80} = -9332.09337289399
x81=7461.43520968971x_{81} = 7461.43520968971
x82=9205.78287224046x_{82} = 9205.78287224046
x83=−438.152083521503x_{83} = -438.152083521503
x84=−840.441569417277x_{84} = -840.441569417277
x85=10732.1431849405x_{85} = 10732.1431849405
x86=−8459.91451555251x_{86} = -8459.91451555251
x87=2665.82801202265x_{87} = 2665.82801202265
x88=5717.21399481699x_{88} = 5717.21399481699
x89=−3228.07574210363x_{89} = -3228.07574210363
x90=1142.88694424986x_{90} = 1142.88694424986
x91=−9550.14106160712x_{91} = -9550.14106160712
x92=−10422.3414304807x_{92} = -10422.3414304807
x93=−9986.23945173436x_{93} = -9986.23945173436
x94=−1270.81536028574x_{94} = -1270.81536028574
x95=−5625.55976637156x_{95} = -5625.55976637156
x96=−4971.57205718097x_{96} = -4971.57205718097
x97=−9114.04678664701x_{97} = -9114.04678664701
x98=−3445.95016102864x_{98} = -3445.95016102864
x99=7897.51346612186x_{99} = 7897.51346612186
x100=−6715.6387454665x_{100} = -6715.6387454665
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^3/(2*(x - 1))^7.
03((−1)2)7\frac{0^{3}}{\left(\left(-1\right) 2\right)^{7}}
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
−7x3128(x−1)8+3x21128(x−1)7=0- \frac{7 x^{3}}{128 \left(x - 1\right)^{8}} + 3 x^{2} \frac{1}{128 \left(x - 1\right)^{7}} = 0
Solve this equation
The roots of this equation
x1=−34x_{1} = - \frac{3}{4}
x2=0x_{2} = 0
The values of the extrema at the points:
         54   
(-3/4, ------)
       823543 

(0, 0)


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:
x2=−34x_{2} = - \frac{3}{4}
Decreasing at intervals
(−∞,−34]\left(-\infty, - \frac{3}{4}\right]
Increasing at intervals
[−34,∞)\left[- \frac{3}{4}, \infty\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
x(28x2(x−1)2−21xx−1+3)64(x−1)7=0\frac{x \left(\frac{28 x^{2}}{\left(x - 1\right)^{2}} - \frac{21 x}{x - 1} + 3\right)}{64 \left(x - 1\right)^{7}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−34−10520x_{2} = - \frac{3}{4} - \frac{\sqrt{105}}{20}
x3=−34+10520x_{3} = - \frac{3}{4} + \frac{\sqrt{105}}{20}
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=1x_{1} = 1

lim⁡x→1−(x(28x2(x−1)2−21xx−1+3)64(x−1)7)=−∞\lim_{x \to 1^-}\left(\frac{x \left(\frac{28 x^{2}}{\left(x - 1\right)^{2}} - \frac{21 x}{x - 1} + 3\right)}{64 \left(x - 1\right)^{7}}\right) = -\infty
lim⁡x→1+(x(28x2(x−1)2−21xx−1+3)64(x−1)7)=∞\lim_{x \to 1^+}\left(\frac{x \left(\frac{28 x^{2}}{\left(x - 1\right)^{2}} - \frac{21 x}{x - 1} + 3\right)}{64 \left(x - 1\right)^{7}}\right) = \infty
- the limits are not equal, so
x1=1x_{1} = 1
- is an inflection point

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
(−∞,−34−10520]∪[−34+10520,∞)\left(-\infty, - \frac{3}{4} - \frac{\sqrt{105}}{20}\right] \cup \left[- \frac{3}{4} + \frac{\sqrt{105}}{20}, \infty\right)
Convex at the intervals
(−∞,−34+10520]∪[0,∞)\left(-\infty, - \frac{3}{4} + \frac{\sqrt{105}}{20}\right] \cup \left[0, \infty\right)
Vertical asymptotes
Have:
x1=1x_{1} = 1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x3(2(x−1))7)=0\lim_{x \to -\infty}\left(\frac{x^{3}}{\left(2 \left(x - 1\right)\right)^{7}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(x3(2(x−1))7)=0\lim_{x \to \infty}\left(\frac{x^{3}}{\left(2 \left(x - 1\right)\right)^{7}}\right) = 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 x^3/(2*(x - 1))^7, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x21128(x−1)7)=0\lim_{x \to -\infty}\left(x^{2} \frac{1}{128 \left(x - 1\right)^{7}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(x21128(x−1)7)=0\lim_{x \to \infty}\left(x^{2} \frac{1}{128 \left(x - 1\right)^{7}}\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:
x3(2(x−1))7=−x3(−2x−2)7\frac{x^{3}}{\left(2 \left(x - 1\right)\right)^{7}} = - \frac{x^{3}}{\left(- 2 x - 2\right)^{7}}
- No
x3(2(x−1))7=x3(−2x−2)7\frac{x^{3}}{\left(2 \left(x - 1\right)\right)^{7}} = \frac{x^{3}}{\left(- 2 x - 2\right)^{7}}
- No
so, the function
not is
neither even, nor odd
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