Mister Exam
Graphing y = x/exp-1/ln(x)
The solution
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x 1
f(x) = -- - ------
x log(x)
e
$$f{\left(x \right)} = \frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}$$
f = x/exp(x) - 1/log(x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1$$
$$x_{1} = 1$$
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:
$$\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- x e^{- x} + \frac{1}{e^{x}} + \frac{1}{x \log{\left(x \right)}^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- x e^{- x} + \frac{1}{e^{x}} + \frac{1}{x \log{\left(x \right)}^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$x e^{- x} - 2 e^{- x} - \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{2} \log{\left(x \right)}^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 92052.0669548957$$
$$x_{2} = 100641.037473986$$
$$x_{3} = 113656.038545734$$
$$x_{4} = 135655.081131858$$
$$x_{5} = 198861.780064415$$
$$x_{6} = 87786.2973905852$$
$$x_{7} = 109301.131390774$$
$$x_{8} = 189709.551623869$$
$$x_{9} = 162484.725665548$$
$$x_{10} = 180594.646831991$$
$$x_{11} = 140096.184290192$$
$$x_{12} = 185147.317387735$$
$$x_{13} = 126812.726236405$$
$$x_{14} = 194281.114117316$$
$$x_{15} = 122412.404500837$$
$$x_{16} = 166996.547043092$$
$$x_{17} = 96337.2699685649$$
$$x_{18} = 203451.334594106$$
$$x_{19} = 104962.570348548$$
$$x_{20} = 118026.659073151$$
$$x_{21} = 153494.18780763$$
$$x_{22} = 157983.833341669$$
$$x_{23} = 131227.111734044$$
$$x_{24} = 149016.124526383$$
$$x_{25} = 144549.998173307$$
$$x_{26} = 171518.996126846$$
$$x_{27} = 176051.78674985$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(x e^{- x} - 2 e^{- x} - \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{2} \log{\left(x \right)}^{3}}\right) = \infty$$
$$\lim_{x \to 1^+}\left(x e^{- x} - 2 e^{- x} - \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{2} \log{\left(x \right)}^{3}}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = 1$$
- is an inflection 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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$x e^{- x} - 2 e^{- x} - \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{2} \log{\left(x \right)}^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 92052.0669548957$$
$$x_{2} = 100641.037473986$$
$$x_{3} = 113656.038545734$$
$$x_{4} = 135655.081131858$$
$$x_{5} = 198861.780064415$$
$$x_{6} = 87786.2973905852$$
$$x_{7} = 109301.131390774$$
$$x_{8} = 189709.551623869$$
$$x_{9} = 162484.725665548$$
$$x_{10} = 180594.646831991$$
$$x_{11} = 140096.184290192$$
$$x_{12} = 185147.317387735$$
$$x_{13} = 126812.726236405$$
$$x_{14} = 194281.114117316$$
$$x_{15} = 122412.404500837$$
$$x_{16} = 166996.547043092$$
$$x_{17} = 96337.2699685649$$
$$x_{18} = 203451.334594106$$
$$x_{19} = 104962.570348548$$
$$x_{20} = 118026.659073151$$
$$x_{21} = 153494.18780763$$
$$x_{22} = 157983.833341669$$
$$x_{23} = 131227.111734044$$
$$x_{24} = 149016.124526383$$
$$x_{25} = 144549.998173307$$
$$x_{26} = 171518.996126846$$
$$x_{27} = 176051.78674985$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(x e^{- x} - 2 e^{- x} - \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{2} \log{\left(x \right)}^{3}}\right) = \infty$$
$$\lim_{x \to 1^+}\left(x e^{- x} - 2 e^{- x} - \frac{1}{x^{2} \log{\left(x \right)}^{2}} - \frac{2}{x^{2} \log{\left(x \right)}^{3}}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = 1$$
- is an inflection 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:
$$x_{1} = 1$$
$$x_{1} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/exp(x) - 1/log(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}}}{x}\right) = 0$$
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:
$$\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}} = - x e^{x} - \frac{1}{\log{\left(- x \right)}}$$
- No
$$\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}} = x e^{x} + \frac{1}{\log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}} = - x e^{x} - \frac{1}{\log{\left(- x \right)}}$$
- No
$$\frac{x}{e^{x}} - \frac{1}{\log{\left(x \right)}} = x e^{x} + \frac{1}{\log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd