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xexp(-x^2/2)
Plotting a function graph/ xexp(-x^2/2)

You entered:

xexp(-x^2/2)

What you mean?

x*exp(-x^2/2)
 
x*exp(-x^1)
 
x*exp(-x^2/2)
 
x*exp(-x*2/2)
 
x*exp(-x*2/2)

Graphing y = xexp(-x^2/2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            2 
          -x  
          ----
           2  
f(x) = x*e
f(x)=xe(−1)x22f{\left(x \right)} = x e^{\frac{\left(-1\right) x^{2}}{2}} 
f = x*exp(-x^2/2)
The graph of the function
0-40-30-20-101020304050601-1
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:
xe(−1)x22=0x e^{\frac{\left(-1\right) x^{2}}{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=9.02289694337605x_{1} = 9.02289694337605
x2=66.5562459808845x_{2} = 66.5562459808845
x3=−40.4883834667848x_{3} = -40.4883834667848
x4=78.5094259967043x_{4} = 78.5094259967043
x5=−28.6943596043799x_{5} = -28.6943596043799
x6=−20.9648808652053x_{6} = -20.9648808652053
x7=−86.228064985273x_{7} = -86.228064985273
x8=−94.2086933283192x_{8} = -94.2086933283192
x9=−84.2334824940775x_{9} = -84.2334824940775
x10=−60.3265078154327x_{10} = -60.3265078154327
x11=−76.2579924560991x_{11} = -76.2579924560991
x12=−74.2649444149412x_{12} = -74.2649444149412
x13=−30.6488081209205x_{13} = -30.6488081209205
x14=94.4654798903143x_{14} = 94.4654798903143
x15=27.0151495573047x_{15} = 27.0151495573047
x16=−62.3160212977879x_{16} = -62.3160212977879
x17=13.8305646213614x_{17} = 13.8305646213614
x18=−72.2722807950237x_{18} = -72.2722807950237
x19=−96.2043534607085x_{19} = -96.2043534607085
x20=−68.2882415884721x_{20} = -68.2882415884721
x21=19.3379596244699x_{21} = 19.3379596244699
x22=−90.2179499984874x_{22} = -90.2179499984874
x23=74.5233591950223x_{23} = 74.5233591950223
x24=90.4750104191475x_{24} = 90.4750104191475
x25=100.45260622169x_{25} = 100.45260622169
x26=15.6247818795334x_{26} = 15.6247818795334
x27=−50.3914159606663x_{27} = -50.3914159606663
x28=92.4701421572582x_{28} = 92.4701421572582
x29=−70.2800343339193x_{29} = -70.2800343339193
x30=46.6877762829739x_{30} = 46.6877762829739
x31=62.5758396530863x_{31} = 62.5758396530863
x32=−19.0686408223086x_{32} = -19.0686408223086
x33=98.4567232107033x_{33} = 98.4567232107033
x34=−88.2228929179074x_{34} = -88.2228929179074
x35=72.5309015703522x_{35} = 72.5309015703522
x36=−82.2391633180928x_{36} = -82.2391633180928
x37=−100.196193168581x_{37} = -100.196193168581
x38=38.7783713446935x_{38} = 38.7783713446935
x39=−48.4076116917235x_{39} = -48.4076116917235
x40=25.0766626196802x_{40} = 25.0766626196802
x41=58.5981051512238x_{41} = 58.5981051512238
x42=60.5866054706585x_{42} = 60.5866054706585
x43=−56.3497113142373x_{43} = -56.3497113142373
x44=−78.2513955147154x_{44} = -78.2513955147154
x45=28.9620890166341x_{45} = 28.9620890166341
x46=−32.6088462722306x_{46} = -32.6088462722306
x47=−46.4252005910588x_{47} = -46.4252005910588
x48=64.5657397621909x_{48} = 64.5657397621909
x49=−8.82065718477295x_{49} = -8.82065718477295
x50=17.465102009593x_{50} = 17.465102009593
x51=42.7288452402976x_{51} = 42.7288452402976
x52=7.83657923190551x_{52} = 7.83657923190551
x53=12.1040591364533x_{53} = 12.1040591364533
x54=23.1487959086869x_{54} = 23.1487959086869
x55=−17.1969684851483x_{55} = -17.1969684851483
x56=−54.362591820007x_{56} = -54.362591820007
x57=48.6697660202548x_{57} = 48.6697660202548
x58=30.9158612891769x_{58} = 30.9158612891769
x59=84.4909972102885x_{59} = 84.4909972102885
x60=−52.3764545425553x_{60} = -52.3764545425553
x61=−34.5735053916839x_{61} = -34.5735053916839
x62=76.5162106782208x_{62} = 76.5162106782208
x63=32.8752332341382x_{63} = 32.8752332341382
x64=54.623627287895x_{64} = 54.623627287895
x65=−58.3377124249816x_{65} = -58.3377124249816
x66=−80.2451271074651x_{66} = -80.2451271074651
x67=−36.5420283825472x_{67} = -36.5420283825472
x68=96.4610108264976x_{68} = 96.4610108264976
x69=21.2345123835376x_{69} = 21.2345123835376
x70=44.7073945322353x_{70} = 44.7073945322353
x71=−7.6886044567655x_{71} = -7.6886044567655
x72=52.6378406446428x_{72} = 52.6378406446428
x73=−38.5138150497611x_{73} = -38.5138150497611
x74=56.6104161729572x_{74} = 56.6104161729572
x75=−15.3593372555059x_{75} = -15.3593372555059
x76=34.8392502079753x_{76} = 34.8392502079753
x77=82.4968425477036x_{77} = 82.4968425477036
x78=−10.247960314217x_{78} = -10.247960314217
x79=88.4800986212492x_{79} = 88.4800986212492
x80=−11.8535751456397x_{80} = -11.8535751456397
x81=68.5473054354995x_{81} = 68.5473054354995
x82=10.4810551180426x_{82} = 10.4810551180426
x83=36.8071620516744x_{83} = 36.8071620516744
x84=−66.2969435075443x_{84} = -66.2969435075443
x85=0x_{85} = 0
x86=−22.8793491121891x_{86} = -22.8793491121891
x87=80.5029780991381x_{87} = 80.5029780991381
x88=−42.4653421912683x_{88} = -42.4653421912683
x89=50.6531744084895x_{89} = 50.6531744084895
x90=−98.2001902620909x_{90} = -98.2001902620909
x91=70.5388712182941x_{91} = 70.5388712182941
x92=−64.3061861108853x_{92} = -64.3061861108853
x93=−13.5703851876112x_{93} = -13.5703851876112
x94=−26.7467597908737x_{94} = -26.7467597908737
x95=−24.8076741592432x_{95} = -24.8076741592432
x96=−92.2132213430942x_{96} = -92.2132213430942
x97=−44.4443699549213x_{97} = -44.4443699549213
x98=40.7523965392307x_{98} = 40.7523965392307
x99=86.4854219951314x_{99} = 86.4854219951314
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*exp(-x^2/2).
0e(−1)0220 e^{\frac{\left(-1\right) 0^{2}}{2}}
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
−x2e(−1)x22+e(−1)x22=0- x^{2} e^{\frac{\left(-1\right) x^{2}}{2}} + e^{\frac{\left(-1\right) x^{2}}{2}} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1
x2=1x_{2} = 1
The values of the extrema at the points:
       -1/2 
(-1, -e    )

     -1/2 
(1, e    )


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=−1x_{1} = -1
Maxima of the function at points:
x1=1x_{1} = 1
Decreasing at intervals
[−1,1]\left[-1, 1\right]
Increasing at intervals
(−∞,−1]∪[1,∞)\left(-\infty, -1\right] \cup \left[1, \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(x2−3)e−x22=0x \left(x^{2} - 3\right) e^{- \frac{x^{2}}{2}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−3x_{2} = - \sqrt{3}
x3=3x_{3} = \sqrt{3}

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
[−3,0]∪[3,∞)\left[- \sqrt{3}, 0\right] \cup \left[\sqrt{3}, \infty\right)
Convex at the intervals
(−∞,−3]∪[0,3]\left(-\infty, - \sqrt{3}\right] \cup \left[0, \sqrt{3}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xe(−1)x22)=0\lim_{x \to -\infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(xe(−1)x22)=0\lim_{x \to \infty}\left(x e^{\frac{\left(-1\right) x^{2}}{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 x*exp(-x^2/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞e(−1)x22=0\lim_{x \to -\infty} e^{\frac{\left(-1\right) x^{2}}{2}} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞e(−1)x22=0\lim_{x \to \infty} e^{\frac{\left(-1\right) x^{2}}{2}} = 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:
xe(−1)x22=−xe(−1)x22x e^{\frac{\left(-1\right) x^{2}}{2}} = - x e^{\frac{\left(-1\right) x^{2}}{2}}
- No
xe(−1)x22=xe(−1)x22x e^{\frac{\left(-1\right) x^{2}}{2}} = x e^{\frac{\left(-1\right) x^{2}}{2}}
- Yes
so, the function
is
odd
The graph
Graphing y = xexp(-x^2/2)
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