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  • Graphing y =:
  • -x^4+2x^2+1
  • x^3/(x-2)^2
  • x^3+x^2-2
  • x^3/(x-1)^2

  • Identical expressions

  • (sqrt(x))*(ln(x))^ two
  • ( square root of (x)) multiply by (ln(x)) squared
  • ( square root of (x)) multiply by (ln(x)) to the power of two
  • (√(x))*(ln(x))^2
  • (sqrt(x))*(ln(x))2
  • sqrtx*lnx2
  • (sqrt(x))*(ln(x))²
  • (sqrt(x))*(ln(x)) to the power of 2
  • (sqrt(x))(ln(x))^2
  • (sqrt(x))(ln(x))2
  • sqrtxlnx2
  • sqrtxlnx^2
Plotting a function graph/ (sqrt(x))*(ln(x))^2

Graphing y = (sqrt(x))*(ln(x))^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
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f(x) = \/ x *log (x)
f(x)=xlog⁡(x)2f{\left(x \right)} = \sqrt{x} \log{\left(x \right)}^{2} 
f = sqrt(x)*log(x)^2
The graph of the function
02468-8-6-4-2-1010020
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:
xlog⁡(x)2=0\sqrt{x} \log{\left(x \right)}^{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=1.00000050151829x_{1} = 1.00000050151829
x2=1.00000087472621x_{2} = 1.00000087472621
x3=1.00000059953375x_{3} = 1.00000059953375
x4=1.00000094198602x_{4} = 1.00000094198602
x5=1.00000090336182x_{5} = 1.00000090336182
x6=1.00000079564216x_{6} = 1.00000079564216
x7=1.00000095363529x_{7} = 1.00000095363529
x8=1.0000003437082x_{8} = 1.0000003437082
x9=1.00000083510986x_{9} = 1.00000083510986
x10=1.00000061472606x_{10} = 1.00000061472606
x11=1.00000075233125x_{11} = 1.00000075233125
x12=0.999999107992762x_{12} = 0.999999107992762
x13=1.00000021457117x_{13} = 1.00000021457117
x14=1.00000083914674x_{14} = 1.00000083914674
x15=1.00000037718347x_{15} = 1.00000037718347
x16=1.00000092200914x_{16} = 1.00000092200914
x17=1.00000088800753x_{17} = 1.00000088800753
x18=0.999999991909142x_{18} = 0.999999991909142
x19=1.0000009256755x_{19} = 1.0000009256755
x20=1.00000094280649x_{20} = 1.00000094280649
x21=1.00000067841741x_{21} = 1.00000067841741
x22=0.999999663646737x_{22} = 0.999999663646737
x23=1.00000095232642x_{23} = 1.00000095232642
x24=1.00000074276506x_{24} = 1.00000074276506
x25=1.00000095642422x_{25} = 1.00000095642422
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(x)*log(x)^2.
0log⁡(0)2\sqrt{0} \log{\left(0 \right)}^{2}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
log⁡(x)22x+2log⁡(x)x=0\frac{\log{\left(x \right)}^{2}}{2 \sqrt{x}} + \frac{2 \log{\left(x \right)}}{\sqrt{x}} = 0
Solve this equation
The roots of this equation
x1=1x_{1} = 1
x2=e−4x_{2} = e^{-4}
The values of the extrema at the points:
(1, 0)

  -4      -2 
(e , 16*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=e−4x_{1} = e^{-4}
Decreasing at intervals
(−∞,e−4]∪[1,∞)\left(-\infty, e^{-4}\right] \cup \left[1, \infty\right)
Increasing at intervals
[e−4,1]\left[e^{-4}, 1\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
2−log⁡(x)24x32=0\frac{2 - \frac{\log{\left(x \right)}^{2}}{4}}{x^{\frac{3}{2}}} = 0
Solve this equation
The roots of this equation
x1=e−22x_{1} = e^{- 2 \sqrt{2}}
x2=e22x_{2} = e^{2 \sqrt{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
[e−22,e22]\left[e^{- 2 \sqrt{2}}, e^{2 \sqrt{2}}\right]
Convex at the intervals
(−∞,e−22]∪[e22,∞)\left(-\infty, e^{- 2 \sqrt{2}}\right] \cup \left[e^{2 \sqrt{2}}, \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→−∞(xlog⁡(x)2)=∞i\lim_{x \to -\infty}\left(\sqrt{x} \log{\left(x \right)}^{2}\right) = \infty i
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(xlog⁡(x)2)=∞\lim_{x \to \infty}\left(\sqrt{x} \log{\left(x \right)}^{2}\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)*log(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(x)2x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}^{2}}{\sqrt{x}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(x)2x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{2}}{\sqrt{x}}\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:
xlog⁡(x)2=−xlog⁡(−x)2\sqrt{x} \log{\left(x \right)}^{2} = \sqrt{- x} \log{\left(- x \right)}^{2}
- No
xlog⁡(x)2=−−xlog⁡(−x)2\sqrt{x} \log{\left(x \right)}^{2} = - \sqrt{- x} \log{\left(- x \right)}^{2}
- No
so, the function
not is
neither even, nor odd
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