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  • How to use it?

  • Graphing y =:
  • x^-6
  • x^3-x^2-x
  • x/(3+x^2)
  • (x+3)/(x+2)^2
  • Derivative of:
  • x^3/(x^2+5) x^3/(x^2+5)
  • Integral of d{x}:
  • x^3/(x^2+5) x^3/(x^2+5)

  • Identical expressions

  • x^ three /(x^ two + five)
  • x cubed divide by (x squared plus 5)
  • x to the power of three divide by (x to the power of two plus five)
  • x3/(x2+5)
  • x3/x2+5
  • x³/(x²+5)
  • x to the power of 3/(x to the power of 2+5)
  • x^3/x^2+5
  • x^3 divide by (x^2+5)

  • Similar expressions

  • x^3/(x^2-5)
Plotting a function graph/ x^3/(x^2+5)

Graphing y = x^3/(x^2+5)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          3  
         x   
f(x) = ------
        2    
       x  + 5
f(x)=x3x2+5f{\left(x \right)} = \frac{x^{3}}{x^{2} + 5} 
f = x^3/(x^2 + 5)
The graph of the function
02468-8-6-4-2-1010-2020
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:
x3x2+5=0\frac{x^{3}}{x^{2} + 5} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=3.45335991975295⋅10−5x_{1} = 3.45335991975295 \cdot 10^{-5}
x2=6.32882439308626⋅10−5x_{2} = 6.32882439308626 \cdot 10^{-5}
x3=−4.31925641118384⋅10−5x_{3} = -4.31925641118384 \cdot 10^{-5}
x4=−4.02078090543322⋅10−5x_{4} = -4.02078090543322 \cdot 10^{-5}
x5=6.81904136080513⋅10−5x_{5} = 6.81904136080513 \cdot 10^{-5}
x6=4.77677969294845⋅10−5x_{6} = 4.77677969294845 \cdot 10^{-5}
x7=3.59587308458976⋅10−5x_{7} = 3.59587308458976 \cdot 10^{-5}
x8=−4.79314545162124⋅10−5x_{8} = -4.79314545162124 \cdot 10^{-5}
x9=−0.000126861661174131x_{9} = -0.000126861661174131
x10=−4.21498802747599⋅10−5x_{10} = -4.21498802747599 \cdot 10^{-5}
x11=4.10351166800191⋅10−5x_{11} = 4.10351166800191 \cdot 10^{-5}
x12=3.67161348782869⋅10−5x_{12} = 3.67161348782869 \cdot 10^{-5}
x13=3.52318181951979⋅10−5x_{13} = 3.52318181951979 \cdot 10^{-5}
x14=−4.92818881318252⋅10−5x_{14} = -4.92818881318252 \cdot 10^{-5}
x15=5.53134119612991⋅10−5x_{15} = 5.53134119612991 \cdot 10^{-5}
x16=−3.843598136445⋅10−5x_{16} = -3.843598136445 \cdot 10^{-5}
x17=−6.35741714538946⋅10−5x_{17} = -6.35741714538946 \cdot 10^{-5}
x18=−9.39219227831315⋅10−5x_{18} = -9.39219227831315 \cdot 10^{-5}
x19=3.83303876752029⋅10−5x_{19} = 3.83303876752029 \cdot 10^{-5}
x20=−6.85217332122653⋅10−5x_{20} = -6.85217332122653 \cdot 10^{-5}
x21=−3.53210776079709⋅10−5x_{21} = -3.53210776079709 \cdot 10^{-5}
x22=5.20285356853379⋅10−5x_{22} = 5.20285356853379 \cdot 10^{-5}
x23=−8.10917310654345⋅10−5x_{23} = -8.10917310654345 \cdot 10^{-5}
x24=−0.000146961292831119x_{24} = -0.000146961292831119
x25=5.71147109324688⋅10−5x_{25} = 5.71147109324688 \cdot 10^{-5}
x26=−0.000173721455793577x_{26} = -0.000173721455793577
x27=4.52924096406739⋅10−5x_{27} = 4.52924096406739 \cdot 10^{-5}
x28=4.41478864122088⋅10−5x_{28} = 4.41478864122088 \cdot 10^{-5}
x29=−3.93020377591954⋅10−5x_{29} = -3.93020377591954 \cdot 10^{-5}
x30=−0.000118635614053435x_{30} = -0.000118635614053435
x31=0x_{31} = 0
x32=−3.46193653675903⋅10−5x_{32} = -3.46193653675903 \cdot 10^{-5}
x33=8.4466165721427⋅10−5x_{33} = 8.4466165721427 \cdot 10^{-5}
x34=−5.07099515749885⋅10−5x_{34} = -5.07099515749885 \cdot 10^{-5}
x35=−8.49707113248183⋅10−5x_{35} = -8.49707113248183 \cdot 10^{-5}
x36=0.000134987053712586x_{36} = 0.000134987053712586
x37=−4.66525342738406⋅10−5x_{37} = -4.66525342738406 \cdot 10^{-5}
x38=−5.22224722930784⋅10−5x_{38} = -5.22224722930784 \cdot 10^{-5}
x39=4.00923231326507⋅10−5x_{39} = 4.00923231326507 \cdot 10^{-5}
x40=8.86738420955485⋅10−5x_{40} = 8.86738420955485 \cdot 10^{-5}
x41=3.91916631341753⋅10−5x_{41} = 3.91916631341753 \cdot 10^{-5}
x42=9.84385697223861⋅10−5x_{42} = 9.84385697223861 \cdot 10^{-5}
x43=4.30594334699575⋅10−5x_{43} = 4.30594334699575 \cdot 10^{-5}
x44=−4.11560752264363⋅10−5x_{44} = -4.11560752264363 \cdot 10^{-5}
x45=−0.000104899396929208x_{45} = -0.000104899396929208
x46=−4.54396324462819⋅10−5x_{46} = -4.54396324462819 \cdot 10^{-5}
x47=3.75059806622827⋅10−5x_{47} = 3.75059806622827 \cdot 10^{-5}
x48=−7.42886944715212⋅10−5x_{48} = -7.42886944715212 \cdot 10^{-5}
x49=7.09326944940362⋅10−5x_{49} = 7.09326944940362 \cdot 10^{-5}
x50=−0.000111364676630913x_{50} = -0.000111364676630913
x51=4.910897042826⋅10−5x_{51} = 4.910897042826 \cdot 10^{-5}
x52=−5.55324007344848⋅10−5x_{52} = -5.55324007344848 \cdot 10^{-5}
x53=0.000171802246191004x_{53} = 0.000171802246191004
x54=−0.000159344965312212x_{54} = -0.000159344965312212
x55=7.39004973612396⋅10−5x_{55} = 7.39004973612396 \cdot 10^{-5}
x56=−0.000136227420056085x_{56} = -0.000136227420056085
x57=7.71222090748369⋅10−5x_{57} = 7.71222090748369 \cdot 10^{-5}
x58=−4.42877991557619⋅10−5x_{58} = -4.42877991557619 \cdot 10^{-5}
x59=−6.59566668076175⋅10−5x_{59} = -6.59566668076175 \cdot 10^{-5}
x60=−9.9118262767697⋅10−5x_{60} = -9.9118262767697 \cdot 10^{-5}
x61=0.000157695734453135x_{61} = 0.000157695734453135
x62=5.05269722903029⋅10−5x_{62} = 5.05269722903029 \cdot 10^{-5}
x63=5.36211958905726⋅10−5x_{63} = 5.36211958905726 \cdot 10^{-5}
x64=−5.92850208459101⋅10−5x_{64} = -5.92850208459101 \cdot 10^{-5}
x65=0.000125775254404067x_{65} = 0.000125775254404067
x66=0.000104141572341571x_{66} = 0.000104141572341571
x67=0.000145535704727759x_{67} = 0.000145535704727759
x68=9.33092623968269⋅10−5x_{68} = 9.33092623968269 \cdot 10^{-5}
x69=6.56492768024551⋅10−5x_{69} = 6.56492768024551 \cdot 10^{-5}
x70=8.06310597124283⋅10−5x_{70} = 8.06310597124283 \cdot 10^{-5}
x71=0.000110515194267213x_{71} = 0.000110515194267213
x72=−7.12907959856879⋅10−5x_{72} = -7.12907959856879 \cdot 10^{-5}
x73=0.000117677889387243x_{73} = 0.000117677889387243
x74=4.20230531171353⋅10−5x_{74} = 4.20230531171353 \cdot 10^{-5}
x75=−8.92286356611081⋅10−5x_{75} = -8.92286356611081 \cdot 10^{-5}
x76=−3.76070960411792⋅10−5x_{76} = -3.76070960411792 \cdot 10^{-5}
x77=6.10890693267222⋅10−5x_{77} = 6.10890693267222 \cdot 10^{-5}
x78=−3.60517003851018⋅10−5x_{78} = -3.60517003851018 \cdot 10^{-5}
x79=5.903585949286⋅10−5x_{79} = 5.903585949286 \cdot 10^{-5}
x80=−5.73480645909681⋅10−5x_{80} = -5.73480645909681 \cdot 10^{-5}
x81=−6.13556782132534⋅10−5x_{81} = -6.13556782132534 \cdot 10^{-5}
x82=−5.38270939611747⋅10−5x_{82} = -5.38270939611747 \cdot 10^{-5}
x83=4.64974176294179⋅10−5x_{83} = 4.64974176294179 \cdot 10^{-5}
x84=−7.75443715160563⋅10−5x_{84} = -7.75443715160563 \cdot 10^{-5}
x85=−3.68130496740476⋅10−5x_{85} = -3.68130496740476 \cdot 10^{-5}
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x^3/(x^2 + 5).
0302+5\frac{0^{3}}{0^{2} + 5}
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
−2x4(x2+5)2+3x2x2+5=0- \frac{2 x^{4}}{\left(x^{2} + 5\right)^{2}} + \frac{3 x^{2}}{x^{2} + 5} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
The values of the extrema at the points:
(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
The function has no maxima
Increasing 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
2x(x2(4x2x2+5−1)x2+5−6x2x2+5+3)x2+5=0\frac{2 x \left(\frac{x^{2} \left(\frac{4 x^{2}}{x^{2} + 5} - 1\right)}{x^{2} + 5} - \frac{6 x^{2}}{x^{2} + 5} + 3\right)}{x^{2} + 5} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−15x_{2} = - \sqrt{15}
x3=15x_{3} = \sqrt{15}

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
(−∞,−15]∪[0,∞)\left(-\infty, - \sqrt{15}\right] \cup \left[0, \infty\right)
Convex at the intervals
(−∞,0]∪[15,∞)\left(-\infty, 0\right] \cup \left[\sqrt{15}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x3x2+5)=−∞\lim_{x \to -\infty}\left(\frac{x^{3}}{x^{2} + 5}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x3x2+5)=∞\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 5}\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^3/(x^2 + 5), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x2x2+5)=1\lim_{x \to -\infty}\left(\frac{x^{2}}{x^{2} + 5}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the left:
y=xy = x
lim⁡x→∞(x2x2+5)=1\lim_{x \to \infty}\left(\frac{x^{2}}{x^{2} + 5}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the right:
y=xy = x
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:
x3x2+5=−x3x2+5\frac{x^{3}}{x^{2} + 5} = - \frac{x^{3}}{x^{2} + 5}
- No
x3x2+5=x3x2+5\frac{x^{3}}{x^{2} + 5} = \frac{x^{3}}{x^{2} + 5}
- Yes
so, the function
is
odd
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