Mister Exam

Other calculators

Plotting a function graph/ sqrt(x)*exp(x)

Graphing y = sqrt(x)*exp(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
         ___  x
f(x) = \/ x *e
f(x)=xexf{\left(x \right)} = \sqrt{x} e^{x} 
f = sqrt(x)*exp(x)
The graph of the function
02468-8-6-4-2-10100100000
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:
xex=0\sqrt{x} e^{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−120.968348080616x_{1} = -120.968348080616
x2=−63.0791364380438x_{2} = -63.0791364380438
x3=−94.9986434020122x_{3} = -94.9986434020122
x4=−100.990069110598x_{4} = -100.990069110598
x5=−98.9927946593729x_{5} = -98.9927946593729
x6=−49.1600217512013x_{6} = -49.1600217512013
x7=−31.4710200385529x_{7} = -31.4710200385529
x8=−53.1309970905921x_{8} = -53.1309970905921
x9=−89.0085679306054x_{9} = -89.0085679306054
x10=−73.0447315518853x_{10} = -73.0447315518853
x11=−61.0877539258535x_{11} = -61.0877539258535
x12=−51.1447223743512x_{12} = -51.1447223743512
x13=−67.0638346231616x_{13} = -67.0638346231616
x14=−96.9956495959489x_{14} = -96.9956495959489
x15=−33.4042694137578x_{15} = -33.4042694137578
x16=−71.0506578790373x_{16} = -71.0506578790373
x17=−69.0570095659301x_{17} = -69.0570095659301
x18=−106.982586137274x_{18} = -106.982586137274
x19=−59.0971307960406x_{19} = -59.0971307960406
x20=−55.1186108210408x_{20} = -55.1186108210408
x21=−112.975997979872x_{21} = -112.975997979872
x22=−91.0050904444827x_{22} = -91.0050904444827
x23=−43.2187501477016x_{23} = -43.2187501477016
x24=−87.01223309669x_{24} = -87.01223309669
x25=−114.973973493728x_{25} = -114.973973493728
x26=0x_{26} = 0
x27=−57.1073737496955x_{27} = -57.1073737496955
x28=−79.0291136628044x_{28} = -79.0291136628044
x29=−45.1966038596489x_{29} = -45.1966038596489
x30=−85.0161016524756x_{30} = -85.0161016524756
x31=−93.0017865135215x_{31} = -93.0017865135215
x32=−77.0339935679106x_{32} = -77.0339935679106
x33=−39.2740530264455x_{33} = -39.2740530264455
x34=−37.3093090541517x_{34} = -37.3093090541517
x35=−83.0201911227922x_{35} = -83.0201911227922
x36=−102.987464315167x_{36} = -102.987464315167
x37=−116.972026550028x_{37} = -116.972026550028
x38=−118.970152765141x_{38} = -118.970152765141
x39=−108.980298944191x_{39} = -108.980298944191
x40=−41.2442746296924x_{40} = -41.2442746296924
x41=−65.0711886544365x_{41} = -65.0711886544365
x42=−81.0245211192852x_{42} = -81.0245211192852
x43=−47.1771902579016x_{43} = -47.1771902579016
x44=−110.978104750617x_{44} = -110.978104750617
x45=−75.0391889040362x_{45} = -75.0391889040362
x46=−104.984972392956x_{46} = -104.984972392956
x47=−35.3518165130442x_{47} = -35.3518165130442
x48=−29.5597318722168x_{48} = -29.5597318722168
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(x)*exp(x).
0e0\sqrt{0} e^{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
xex+ex2x=0\sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}} = 0
Solve this equation
The roots of this equation
x1=−12x_{1} = - \frac{1}{2}
The values of the extrema at the points:
           ___  -1/2 
       I*\/ 2 *e     
(-1/2, -------------)
             2


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
Doesn't change the value 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
(x+1x−14x32)ex=0\left(\sqrt{x} + \frac{1}{\sqrt{x}} - \frac{1}{4 x^{\frac{3}{2}}}\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=−12+22x_{1} = - \frac{1}{2} + \frac{\sqrt{2}}{2}
x2=−22−12x_{2} = - \frac{\sqrt{2}}{2} - \frac{1}{2}

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
[−12+22,∞)\left[- \frac{1}{2} + \frac{\sqrt{2}}{2}, \infty\right)
Convex at the intervals
(−∞,−12+22]\left(-\infty, - \frac{1}{2} + \frac{\sqrt{2}}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xex)=0\lim_{x \to -\infty}\left(\sqrt{x} e^{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(xex)=∞\lim_{x \to \infty}\left(\sqrt{x} e^{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 sqrt(x)*exp(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(exx)=0\lim_{x \to -\infty}\left(\frac{e^{x}}{\sqrt{x}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(exx)=∞\lim_{x \to \infty}\left(\frac{e^{x}}{\sqrt{x}}\right) = \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:
xex=−xe−x\sqrt{x} e^{x} = \sqrt{- x} e^{- x}
- No
xex=−−xe−x\sqrt{x} e^{x} = - \sqrt{- x} e^{- x}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator