Mister Exam
Graphing y = exp(x)/(1-x*exp(x)+exp(x))
The solution
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x
e
f(x) = -------------
x x
1 - x*e + e
$$f{\left(x \right)} = \frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}}$$
f = exp(x)/(-x*exp(x) + 1 + exp(x))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1.27846454276107$$
$$x_{1} = 1.27846454276107$$
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:
$$\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{x e^{2 x}}{\left(\left(- x e^{x} + 1\right) + e^{x}\right)^{2}} + \frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}} = 0$$
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
$$\frac{x e^{2 x}}{\left(\left(- x e^{x} + 1\right) + e^{x}\right)^{2}} + \frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}} = 0$$
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
$$\frac{\left(\frac{2 x e^{x}}{- x e^{x} + e^{x} + 1} + 1 + \frac{\left(\frac{2 x^{2} e^{x}}{- x e^{x} + e^{x} + 1} + x + 1\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) e^{x}}{- x e^{x} + e^{x} + 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 15747.4072323157$$
$$x_{2} = 38632.655060232$$
$$x_{3} = 39480.2583130978$$
$$x_{4} = 36089.845578694$$
$$x_{5} = -114.872003083$$
$$x_{6} = 30156.6255709251$$
$$x_{7} = -84.8720030830002$$
$$x_{8} = 37785.0518514713$$
$$x_{9} = 13204.6161538175$$
$$x_{10} = -118.872003083$$
$$x_{11} = 33547.0365862417$$
$$x_{12} = -36.8720157544651$$
$$x_{13} = -40.8720033751864$$
$$x_{14} = 25918.6139827621$$
$$x_{15} = -108.872003083$$
$$x_{16} = -94.8720030830002$$
$$x_{17} = -96.8720030830002$$
$$x_{18} = -34.8720845904371$$
$$x_{19} = 41175.4649402498$$
$$x_{20} = -80.8720030830002$$
$$x_{21} = -74.8720030830002$$
$$x_{22} = -70.8720030830002$$
$$x_{23} = 29309.0230313971$$
$$x_{24} = 14899.80938599$$
$$x_{25} = 14052.2123103628$$
$$x_{26} = 34394.6395226874$$
$$x_{27} = -62.8720030830002$$
$$x_{28} = -104.872003083$$
$$x_{29} = 21680.6061441251$$
$$x_{30} = -72.8720030830002$$
$$x_{31} = -38.8720050199378$$
$$x_{32} = 12357.0211054262$$
$$x_{33} = -32.8725157813207$$
$$x_{34} = 36937.4486898515$$
$$x_{35} = 23375.8086991307$$
$$x_{36} = -106.872003083$$
$$x_{37} = -46.8720030839479$$
$$x_{38} = -78.8720030830002$$
$$x_{39} = -56.8720030830003$$
$$x_{40} = 19137.8042768329$$
$$x_{41} = 31004.2282032034$$
$$x_{42} = 16595.0057312651$$
$$x_{43} = -54.8720030830006$$
$$x_{44} = -60.8720030830002$$
$$x_{45} = -50.8720030830203$$
$$x_{46} = 24223.4102871667$$
$$x_{47} = -58.8720030830002$$
$$x_{48} = -116.872003083$$
$$x_{49} = 17442.6047877107$$
$$x_{50} = -90.8720030830002$$
$$x_{51} = -30.8751299511315$$
$$x_{52} = -64.8720030830002$$
$$x_{53} = -28.8902259413341$$
$$x_{54} = 19985.4045927882$$
$$x_{55} = -82.8720030830002$$
$$x_{56} = -110.872003083$$
$$x_{57} = -42.8720031266101$$
$$x_{58} = -92.8720030830002$$
$$x_{59} = 40327.8616072876$$
$$x_{60} = 26766.2160574209$$
$$x_{61} = -44.8720030894525$$
$$x_{62} = 20833.0052276867$$
$$x_{63} = 18290.2043241563$$
$$x_{64} = 31851.830920828$$
$$x_{65} = -88.8720030830002$$
$$x_{66} = -112.872003083$$
$$x_{67} = -48.8720030831385$$
$$x_{68} = 27613.8182647528$$
$$x_{69} = -100.872003083$$
$$x_{70} = -68.8720030830002$$
$$x_{71} = -120.872003083$$
$$x_{72} = 35242.2425216394$$
$$x_{73} = -102.872003083$$
$$x_{74} = -86.8720030830002$$
$$x_{75} = -66.8720030830002$$
$$x_{76} = -76.8720030830002$$
$$x_{77} = 42023.0683096383$$
$$x_{78} = -98.8720030830002$$
$$x_{79} = 25071.0120542313$$
$$x_{80} = 22528.2073103286$$
$$x_{81} = 28461.4205929053$$
$$x_{82} = -52.8720030830031$$
$$x_{83} = 32699.4337171627$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1.27846454276107$$
$$\lim_{x \to 1.27846454276107^-}\left(\frac{\left(\frac{2 x e^{x}}{- x e^{x} + e^{x} + 1} + 1 + \frac{\left(\frac{2 x^{2} e^{x}}{- x e^{x} + e^{x} + 1} + x + 1\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) = -2.16071262422532 \cdot 10^{47}$$
$$\lim_{x \to 1.27846454276107^+}\left(\frac{\left(\frac{2 x e^{x}}{- x e^{x} + e^{x} + 1} + 1 + \frac{\left(\frac{2 x^{2} e^{x}}{- x e^{x} + e^{x} + 1} + x + 1\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) = -2.16071262422532 \cdot 10^{47}$$
- limits are equal, then skip the corresponding 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
$$\frac{\left(\frac{2 x e^{x}}{- x e^{x} + e^{x} + 1} + 1 + \frac{\left(\frac{2 x^{2} e^{x}}{- x e^{x} + e^{x} + 1} + x + 1\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) e^{x}}{- x e^{x} + e^{x} + 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 15747.4072323157$$
$$x_{2} = 38632.655060232$$
$$x_{3} = 39480.2583130978$$
$$x_{4} = 36089.845578694$$
$$x_{5} = -114.872003083$$
$$x_{6} = 30156.6255709251$$
$$x_{7} = -84.8720030830002$$
$$x_{8} = 37785.0518514713$$
$$x_{9} = 13204.6161538175$$
$$x_{10} = -118.872003083$$
$$x_{11} = 33547.0365862417$$
$$x_{12} = -36.8720157544651$$
$$x_{13} = -40.8720033751864$$
$$x_{14} = 25918.6139827621$$
$$x_{15} = -108.872003083$$
$$x_{16} = -94.8720030830002$$
$$x_{17} = -96.8720030830002$$
$$x_{18} = -34.8720845904371$$
$$x_{19} = 41175.4649402498$$
$$x_{20} = -80.8720030830002$$
$$x_{21} = -74.8720030830002$$
$$x_{22} = -70.8720030830002$$
$$x_{23} = 29309.0230313971$$
$$x_{24} = 14899.80938599$$
$$x_{25} = 14052.2123103628$$
$$x_{26} = 34394.6395226874$$
$$x_{27} = -62.8720030830002$$
$$x_{28} = -104.872003083$$
$$x_{29} = 21680.6061441251$$
$$x_{30} = -72.8720030830002$$
$$x_{31} = -38.8720050199378$$
$$x_{32} = 12357.0211054262$$
$$x_{33} = -32.8725157813207$$
$$x_{34} = 36937.4486898515$$
$$x_{35} = 23375.8086991307$$
$$x_{36} = -106.872003083$$
$$x_{37} = -46.8720030839479$$
$$x_{38} = -78.8720030830002$$
$$x_{39} = -56.8720030830003$$
$$x_{40} = 19137.8042768329$$
$$x_{41} = 31004.2282032034$$
$$x_{42} = 16595.0057312651$$
$$x_{43} = -54.8720030830006$$
$$x_{44} = -60.8720030830002$$
$$x_{45} = -50.8720030830203$$
$$x_{46} = 24223.4102871667$$
$$x_{47} = -58.8720030830002$$
$$x_{48} = -116.872003083$$
$$x_{49} = 17442.6047877107$$
$$x_{50} = -90.8720030830002$$
$$x_{51} = -30.8751299511315$$
$$x_{52} = -64.8720030830002$$
$$x_{53} = -28.8902259413341$$
$$x_{54} = 19985.4045927882$$
$$x_{55} = -82.8720030830002$$
$$x_{56} = -110.872003083$$
$$x_{57} = -42.8720031266101$$
$$x_{58} = -92.8720030830002$$
$$x_{59} = 40327.8616072876$$
$$x_{60} = 26766.2160574209$$
$$x_{61} = -44.8720030894525$$
$$x_{62} = 20833.0052276867$$
$$x_{63} = 18290.2043241563$$
$$x_{64} = 31851.830920828$$
$$x_{65} = -88.8720030830002$$
$$x_{66} = -112.872003083$$
$$x_{67} = -48.8720030831385$$
$$x_{68} = 27613.8182647528$$
$$x_{69} = -100.872003083$$
$$x_{70} = -68.8720030830002$$
$$x_{71} = -120.872003083$$
$$x_{72} = 35242.2425216394$$
$$x_{73} = -102.872003083$$
$$x_{74} = -86.8720030830002$$
$$x_{75} = -66.8720030830002$$
$$x_{76} = -76.8720030830002$$
$$x_{77} = 42023.0683096383$$
$$x_{78} = -98.8720030830002$$
$$x_{79} = 25071.0120542313$$
$$x_{80} = 22528.2073103286$$
$$x_{81} = 28461.4205929053$$
$$x_{82} = -52.8720030830031$$
$$x_{83} = 32699.4337171627$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1.27846454276107$$
$$\lim_{x \to 1.27846454276107^-}\left(\frac{\left(\frac{2 x e^{x}}{- x e^{x} + e^{x} + 1} + 1 + \frac{\left(\frac{2 x^{2} e^{x}}{- x e^{x} + e^{x} + 1} + x + 1\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) = -2.16071262422532 \cdot 10^{47}$$
$$\lim_{x \to 1.27846454276107^+}\left(\frac{\left(\frac{2 x e^{x}}{- x e^{x} + e^{x} + 1} + 1 + \frac{\left(\frac{2 x^{2} e^{x}}{- x e^{x} + e^{x} + 1} + x + 1\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) e^{x}}{- x e^{x} + e^{x} + 1}\right) = -2.16071262422532 \cdot 10^{47}$$
- limits are equal, then skip the corresponding 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:
$$x_{1} = 1.27846454276107$$
$$x_{1} = 1.27846454276107$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}}\right) = 0$$
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 exp(x)/(1 - x*exp(x) + exp(x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{x}}{x \left(\left(- x e^{x} + 1\right) + e^{x}\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x \left(\left(- x e^{x} + 1\right) + e^{x}\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{e^{x}}{x \left(\left(- x e^{x} + 1\right) + e^{x}\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x \left(\left(- x e^{x} + 1\right) + e^{x}\right)}\right) = 0$$
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:
$$\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}} = \frac{e^{- x}}{x e^{- x} + 1 + e^{- x}}$$
- No
$$\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}} = - \frac{e^{- x}}{x e^{- x} + 1 + e^{- x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}} = \frac{e^{- x}}{x e^{- x} + 1 + e^{- x}}$$
- No
$$\frac{e^{x}}{\left(- x e^{x} + 1\right) + e^{x}} = - \frac{e^{- x}}{x e^{- x} + 1 + e^{- x}}$$
- No
so, the function
not is
neither even, nor odd