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  • Graphing y =:
  • -x^3-3x-2
  • x^3+3x^2+5
  • x^3-3x-20
  • x^3+2x^2-7x+4

  • Identical expressions

  • (log(three *x+ two))/x^ two
  • ( logarithm of (3 multiply by x plus 2)) divide by x squared
  • ( logarithm of (three multiply by x plus two)) divide by x to the power of two
  • (log(3*x+2))/x2
  • log3*x+2/x2
  • (log(3*x+2))/x²
  • (log(3*x+2))/x to the power of 2
  • (log(3x+2))/x^2
  • (log(3x+2))/x2
  • log3x+2/x2
  • log3x+2/x^2
  • (log(3*x+2)) divide by x^2

  • Similar expressions

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

Graphing y = (log(3*x+2))/x^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       log(3*x + 2)
f(x) = ------------
             2     
            x
f(x)=log⁡(3x+2)x2f{\left(x \right)} = \frac{\log{\left(3 x + 2 \right)}}{x^{2}} 
f = log(3*x + 2)/x^2
The graph of the function
02468-8-6-4-2-1010-500500
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
log⁡(3x+2)x2=0\frac{\log{\left(3 x + 2 \right)}}{x^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−13x_{1} = - \frac{1}{3}
Numerical solution
x1=−0.333333333333333x_{1} = -0.333333333333333
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(3*x + 2)/x^2.
log⁡(0⋅3+2)02\frac{\log{\left(0 \cdot 3 + 2 \right)}}{0^{2}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
3x2(3x+2)−2log⁡(3x+2)x3=0\frac{3}{x^{2} \left(3 x + 2\right)} - \frac{2 \log{\left(3 x + 2 \right)}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=31713.7126029775x_{1} = 31713.7126029775
x2=26449.1246932315x_{2} = 26449.1246932315
x3=39045.1425615862x_{3} = 39045.1425615862
x4=49458.7397957045x_{4} = 49458.7397957045
x5=53608.4099408452x_{5} = 53608.4099408452
x6=43218.0190536496x_{6} = 43218.0190536496
x7=34860.7140579662x_{7} = 34860.7140579662
x8=40089.3824052297x_{8} = 40089.3824052297
x9=30662.8716201751x_{9} = 30662.8716201751
x10=27504.2211025408x_{10} = 27504.2211025408
x11=32763.607291801x_{11} = 32763.607291801
x12=45300.5879055141x_{12} = 45300.5879055141
x13=51534.5787252644x_{13} = 51534.5787252644
x14=29611.0412564979x_{14} = 29611.0412564979
x15=46340.9735366077x_{15} = 46340.9735366077
x16=35907.9972351282x_{16} = 35907.9972351282
x17=33812.5954880903x_{17} = 33812.5954880903
x18=52571.7388356401x_{18} = 52571.7388356401
x19=42175.7983619252x_{19} = 42175.7983619252
x20=28558.174732813x_{20} = 28558.174732813
x21=44259.6091491924x_{21} = 44259.6091491924
x22=38000.1827304015x_{22} = 38000.1827304015
x23=36954.4768841542x_{23} = 36954.4768841542
x24=48420.0335351419x_{24} = 48420.0335351419
x25=47380.783293281x_{25} = 47380.783293281
x26=41132.9266854406x_{26} = 41132.9266854406
x27=50496.9168414124x_{27} = 50496.9168414124
The values of the extrema at the points:
(31713.712602977546, 1.13974931243923e-8)

(26449.12469323154, 1.61268185252232e-8)

(39045.14256158622, 7.65557614620553e-9)

(49458.739795704525, 4.86782795728042e-9)

(53608.40994084518, 4.17142074368903e-9)

(43218.01905364956, 6.30295446019717e-9)

(34860.71405796623, 9.51044053345461e-9)

(40089.382405229735, 7.27837056780266e-9)

(30662.87162017514, 1.21562429898751e-8)

(27504.221102540836, 1.49649676039635e-8)

(32763.607291801, 1.07090817860131e-8)

(45300.587905514134, 5.75968671314715e-9)

(51534.57872526436, 4.49904959341657e-9)

(29611.04125649793, 1.29953906436391e-8)

(46340.97353660767, 5.51454557922146e-9)

(35907.997235128154, 8.98672837798586e-9)

(33812.595488090294, 1.00824851044884e-8)

(52571.738835640055, 4.33049138857726e-9)

(42175.79836192517, 6.60458924917291e-9)

(28558.174732813015, 1.39268772418209e-8)

(44259.609149192416, 6.02193927024416e-9)

(38000.18273040146, 8.06361779489966e-9)

(36954.476884154166, 8.50599636289094e-9)

(48420.03353514192, 5.06986426400606e-9)

(47380.783293281034, 5.28504353482493e-9)

(41132.92668544059, 6.92893809481599e-9)

(50496.9168414124, 4.67787459760698e-9)


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
Decreasing 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
3(−3(3x+2)2−4x(3x+2)+2log⁡(3x+2)x2)x2=0\frac{3 \left(- \frac{3}{\left(3 x + 2\right)^{2}} - \frac{4}{x \left(3 x + 2\right)} + \frac{2 \log{\left(3 x + 2 \right)}}{x^{2}}\right)}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=8274.96641346692x_{1} = 8274.96641346692
x2=3467.93082080922x_{2} = 3467.93082080922
x3=7269.49494353544x_{3} = 7269.49494353544
x4=6261.24163199901x_{4} = 6261.24163199901
x5=4996.04544900762x_{5} = 4996.04544900762
x6=11028.8868095822x_{6} = 11028.8868095822
x7=7017.71285043971x_{7} = 7017.71285043971
x8=8023.83295673205x_{8} = 8023.83295673205
x9=6513.59565486163x_{9} = 6513.59565486163
x10=9528.55389092141x_{10} = 9528.55389092141
x11=7521.10400316657x_{11} = 7521.10400316657
x12=5755.88834447734x_{12} = 5755.88834447734
x13=9027.51417074017x_{13} = 9027.51417074017
x14=10279.2145372991x_{14} = 10279.2145372991
x15=3978.78874524551x_{15} = 3978.78874524551
x16=7772.54765976361x_{16} = 7772.54765976361
x17=12774.7423712679x_{17} = 12774.7423712679
x18=8525.95407981471x_{18} = 8525.95407981471
x19=5502.86388330909x_{19} = 5502.86388330909
x20=6765.74943313631x_{20} = 6765.74943313631
x21=12027.0605415605x_{21} = 12027.0605415605
x22=6008.67651468855x_{22} = 6008.67651468855
x23=11777.6575438605x_{23} = 11777.6575438605
x24=4742.21649407986x_{24} = 4742.21649407986
x25=10779.1002162637x_{25} = 10779.1002162637
x26=10529.2105261227x_{26} = 10529.2105261227
x27=10029.1088631004x_{27} = 10029.1088631004
x28=4233.61381907963x_{28} = 4233.61381907963
x29=3211.81641803675x_{29} = 3211.81641803675
x30=2955.17712719642x_{30} = 2955.17712719642
x31=4488.0805821838x_{31} = 4488.0805821838
x32=11278.5733373327x_{32} = 11278.5733373327
x33=9778.88991618375x_{33} = 9778.88991618375
x34=12525.600661984x_{34} = 12525.600661984
x35=8776.80158337075x_{35} = 8776.80158337075
x36=2697.94797882468x_{36} = 2697.94797882468
x37=12276.3741315866x_{37} = 12276.3741315866
x38=3723.57351650685x_{38} = 3723.57351650685
x39=11528.1626729929x_{39} = 11528.1626729929
x40=9278.0967436856x_{40} = 9278.0967436856
x41=5249.58840638645x_{41} = 5249.58840638645
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=0x_{1} = 0

lim⁡x→0−(3(−3(3x+2)2−4x(3x+2)+2log⁡(3x+2)x2)x2)=∞\lim_{x \to 0^-}\left(\frac{3 \left(- \frac{3}{\left(3 x + 2\right)^{2}} - \frac{4}{x \left(3 x + 2\right)} + \frac{2 \log{\left(3 x + 2 \right)}}{x^{2}}\right)}{x^{2}}\right) = \infty
lim⁡x→0+(3(−3(3x+2)2−4x(3x+2)+2log⁡(3x+2)x2)x2)=∞\lim_{x \to 0^+}\left(\frac{3 \left(- \frac{3}{\left(3 x + 2\right)^{2}} - \frac{4}{x \left(3 x + 2\right)} + \frac{2 \log{\left(3 x + 2 \right)}}{x^{2}}\right)}{x^{2}}\right) = \infty
- limits are equal, then skip the corresponding 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(log⁡(3x+2)x2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(3 x + 2 \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(log⁡(3x+2)x2)=0\lim_{x \to \infty}\left(\frac{\log{\left(3 x + 2 \right)}}{x^{2}}\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 log(3*x + 2)/x^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(3x+2)xx2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(3 x + 2 \right)}}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(3x+2)xx2)=0\lim_{x \to \infty}\left(\frac{\log{\left(3 x + 2 \right)}}{x x^{2}}\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:
log⁡(3x+2)x2=log⁡(2−3x)x2\frac{\log{\left(3 x + 2 \right)}}{x^{2}} = \frac{\log{\left(2 - 3 x \right)}}{x^{2}}
- No
log⁡(3x+2)x2=−log⁡(2−3x)x2\frac{\log{\left(3 x + 2 \right)}}{x^{2}} = - \frac{\log{\left(2 - 3 x \right)}}{x^{2}}
- No
so, the function
not is
neither even, nor odd
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