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x*3^x
Plotting a function graph/ x*3^x

Graphing y = x*3^x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          x
f(x) = x*3
f(x)=3xxf{\left(x \right)} = 3^{x} x 
f = 3^x*x
The graph of the function
0-100-90-80-70-60-50-40-30-20-1010-5000001000000
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:
3xx=03^{x} x = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−37.6781436701552x_{1} = -37.6781436701552
x2=−107.167274596856x_{2} = -107.167274596856
x3=−49.4505421027084x_{3} = -49.4505421027084
x4=−63.3228960049139x_{4} = -63.3228960049139
x5=−61.3366222633619x_{5} = -61.3366222633619
x6=−85.2216567655223x_{6} = -85.2216567655223
x7=−51.426609895199x_{7} = -51.426609895199
x8=−67.2984527302786x_{8} = -67.2984527302786
x9=−41.5803840500915x_{9} = -41.5803840500915
x10=−77.2505363225033x_{10} = -77.2505363225033
x11=−99.1838768826625x_{11} = -99.1838768826625
x12=−65.3102111808443x_{12} = -65.3102111808443
x13=−71.2773341579085x_{13} = -71.2773341579085
x14=−101.179447462537x_{14} = -101.179447462537
x15=−91.2038299641473x_{15} = -91.2038299641473
x16=−39.6254287506401x_{16} = -39.6254287506401
x17=−117.150067655572x_{17} = -117.150067655572
x18=−115.153242753328x_{18} = -115.153242753328
x19=−119.14701079529x_{19} = -119.14701079529
x20=−111.159976271481x_{20} = -111.159976271481
x21=−35.7407684061747x_{21} = -35.7407684061747
x22=−59.3515251017883x_{22} = -59.3515251017883
x23=−69.287522054867x_{23} = -69.287522054867
x24=−79.2426704800528x_{24} = -79.2426704800528
x25=−73.2678153634158x_{25} = -73.2678153634158
x26=−30.0305613589326x_{26} = -30.0305613589326
x27=0x_{27} = 0
x28=−93.198473257723x_{28} = -93.198473257723
x29=−53.4050531471438x_{29} = -53.4050531471438
x30=−31.9104759351313x_{30} = -31.9104759351313
x31=−47.4772732671719x_{31} = -47.4772732671719
x32=−89.2094642193927x_{32} = -89.2094642193927
x33=−28.1910379196607x_{33} = -28.1910379196607
x34=−87.2153982580997x_{34} = -87.2153982580997
x35=−83.2282672262855x_{35} = -83.2282672262855
x36=−105.171158304958x_{36} = -105.171158304958
x37=−57.3677643839735x_{37} = -57.3677643839735
x38=−109.163550472496x_{38} = -109.163550472496
x39=−97.1885140620293x_{39} = -97.1885140620293
x40=−33.8165589494197x_{40} = -33.8165589494197
x41=−45.5073361732256x_{41} = -45.5073361732256
x42=−75.2589014817435x_{42} = -75.2589014817435
x43=−43.5414114397842x_{43} = -43.5414114397842
x44=−113.156543099422x_{44} = -113.156543099422
x45=−81.2352603347398x_{45} = -81.2352603347398
x46=−55.385530685981x_{46} = -55.385530685981
x47=−95.1933740183726x_{47} = -95.1933740183726
x48=−103.175212107713x_{48} = -103.175212107713
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.
0⋅300 \cdot 3^{0}
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
3xxlog⁡(3)+3x=03^{x} x \log{\left(3 \right)} + 3^{x} = 0
Solve this equation
The roots of this equation
x1=−1log⁡(3)x_{1} = - \frac{1}{\log{\left(3 \right)}}
The values of the extrema at the points:
           -1   
  -1     -e     
(------, ------)
 log(3)  log(3)


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=−1log⁡(3)x_{1} = - \frac{1}{\log{\left(3 \right)}}
The function has no maxima
Decreasing at intervals
[−1log⁡(3),∞)\left[- \frac{1}{\log{\left(3 \right)}}, \infty\right)
Increasing at intervals
(−∞,−1log⁡(3)]\left(-\infty, - \frac{1}{\log{\left(3 \right)}}\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
3x(xlog⁡(3)+2)log⁡(3)=03^{x} \left(x \log{\left(3 \right)} + 2\right) \log{\left(3 \right)} = 0
Solve this equation
The roots of this equation
x1=−2log⁡(3)x_{1} = - \frac{2}{\log{\left(3 \right)}}

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
[−2log⁡(3),∞)\left[- \frac{2}{\log{\left(3 \right)}}, \infty\right)
Convex at the intervals
(−∞,−2log⁡(3)]\left(-\infty, - \frac{2}{\log{\left(3 \right)}}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(3xx)=0\lim_{x \to -\infty}\left(3^{x} x\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(3xx)=∞\lim_{x \to \infty}\left(3^{x} 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*3^x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞3x=0\lim_{x \to -\infty} 3^{x} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞3x=∞\lim_{x \to \infty} 3^{x} = \infty
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
3xx=−3−xx3^{x} x = - 3^{- x} x
- No
3xx=3−xx3^{x} x = 3^{- x} x
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = x*3^x
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