Mister Exam
Graphing y = (x^(2)+2*x-3-x*e^(1/x))/x
The solution
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2 x ___
x + 2*x - 3 - x*\/ E
f(x) = ----------------------
x
$$f{\left(x \right)} = \frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x}$$
f = (-E^(1/x)*x + x^2 + 2*x - 3)/x
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1.65049637905197$$
$$x_{2} = -2.51874872478681$$
so we need to solve the equation:
$$\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1.65049637905197$$
$$x_{2} = -2.51874872478681$$
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
$$\frac{2 x - e^{\frac{1}{x}} + 2 + \frac{e^{\frac{1}{x}}}{x}}{x} - \frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x^{2}} = 0$$
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
$$\frac{2 x - e^{\frac{1}{x}} + 2 + \frac{e^{\frac{1}{x}}}{x}}{x} - \frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x^{2}} = 0$$
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
$$\frac{2 - \frac{2 \left(2 x - e^{\frac{1}{x}} + 2 + \frac{e^{\frac{1}{x}}}{x}\right)}{x} - \frac{2 \left(- x^{2} + x e^{\frac{1}{x}} - 2 x + 3\right)}{x^{2}} - \frac{e^{\frac{1}{x}}}{x^{3}}}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 39002.3332018796$$
$$x_{2} = 19507.2085828027$$
$$x_{3} = 38154.723909211$$
$$x_{4} = -30500.6939976197$$
$$x_{5} = -29653.081395909$$
$$x_{6} = 27983.38854628$$
$$x_{7} = -32195.9178778353$$
$$x_{8} = 37307.1143870768$$
$$x_{9} = 41545.1598433672$$
$$x_{10} = 27135.7744109392$$
$$x_{11} = 22897.6927654468$$
$$x_{12} = -22872.158219297$$
$$x_{13} = -35586.361228948$$
$$x_{14} = 28831.0021005822$$
$$x_{15} = -33043.5292243101$$
$$x_{16} = 25440.5441653164$$
$$x_{17} = 34764.2842762836$$
$$x_{18} = -39824.4090880006$$
$$x_{19} = 29678.6151235411$$
$$x_{20} = -23719.7764131599$$
$$x_{21} = -18634.0481606355$$
$$x_{22} = -37281.5811012877$$
$$x_{23} = -34738.7508750124$$
$$x_{24} = -42367.235238804$$
$$x_{25} = -20329.2966399396$$
$$x_{26} = -27110.2404471593$$
$$x_{27} = 22050.0736537062$$
$$x_{28} = -28805.4683011404$$
$$x_{29} = 24592.9279191901$$
$$x_{30} = -36433.9712978862$$
$$x_{31} = -22024.5389490713$$
$$x_{32} = -27957.8546684026$$
$$x_{33} = 18659.5837299829$$
$$x_{34} = -24567.3936422505$$
$$x_{35} = 36459.5046194954$$
$$x_{36} = 33069.0627178099$$
$$x_{37} = 26288.1596384684$$
$$x_{38} = -26262.6255803404$$
$$x_{39} = -33891.1402146696$$
$$x_{40} = -31348.3061463201$$
$$x_{41} = -38976.79998085$$
$$x_{42} = -38129.1906568808$$
$$x_{43} = 31373.8397474042$$
$$x_{44} = 33916.6736603263$$
$$x_{45} = -21176.9184729939$$
$$x_{46} = 23745.3108174977$$
$$x_{47} = 30526.2276593423$$
$$x_{48} = 40697.5511560972$$
$$x_{49} = 39849.942279706$$
$$x_{50} = 20354.8317230213$$
$$x_{51} = 21202.4533555068$$
$$x_{52} = -25415.0100032502$$
$$x_{53} = 35611.894588966$$
$$x_{54} = -41519.6267050161$$
$$x_{55} = 42392.7683528465$$
$$x_{56} = 32221.4514230046$$
$$x_{57} = -40672.0179919024$$
$$x_{58} = -19481.6732724314$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 - \frac{2 \left(2 x - e^{\frac{1}{x}} + 2 + \frac{e^{\frac{1}{x}}}{x}\right)}{x} - \frac{2 \left(- x^{2} + x e^{\frac{1}{x}} - 2 x + 3\right)}{x^{2}} - \frac{e^{\frac{1}{x}}}{x^{3}}}{x}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{2 - \frac{2 \left(2 x - e^{\frac{1}{x}} + 2 + \frac{e^{\frac{1}{x}}}{x}\right)}{x} - \frac{2 \left(- x^{2} + x e^{\frac{1}{x}} - 2 x + 3\right)}{x^{2}} - \frac{e^{\frac{1}{x}}}{x^{3}}}{x}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- 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:
Concave at the intervals
$$\left[39849.942279706, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -42367.235238804\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 - \frac{2 \left(2 x - e^{\frac{1}{x}} + 2 + \frac{e^{\frac{1}{x}}}{x}\right)}{x} - \frac{2 \left(- x^{2} + x e^{\frac{1}{x}} - 2 x + 3\right)}{x^{2}} - \frac{e^{\frac{1}{x}}}{x^{3}}}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 39002.3332018796$$
$$x_{2} = 19507.2085828027$$
$$x_{3} = 38154.723909211$$
$$x_{4} = -30500.6939976197$$
$$x_{5} = -29653.081395909$$
$$x_{6} = 27983.38854628$$
$$x_{7} = -32195.9178778353$$
$$x_{8} = 37307.1143870768$$
$$x_{9} = 41545.1598433672$$
$$x_{10} = 27135.7744109392$$
$$x_{11} = 22897.6927654468$$
$$x_{12} = -22872.158219297$$
$$x_{13} = -35586.361228948$$
$$x_{14} = 28831.0021005822$$
$$x_{15} = -33043.5292243101$$
$$x_{16} = 25440.5441653164$$
$$x_{17} = 34764.2842762836$$
$$x_{18} = -39824.4090880006$$
$$x_{19} = 29678.6151235411$$
$$x_{20} = -23719.7764131599$$
$$x_{21} = -18634.0481606355$$
$$x_{22} = -37281.5811012877$$
$$x_{23} = -34738.7508750124$$
$$x_{24} = -42367.235238804$$
$$x_{25} = -20329.2966399396$$
$$x_{26} = -27110.2404471593$$
$$x_{27} = 22050.0736537062$$
$$x_{28} = -28805.4683011404$$
$$x_{29} = 24592.9279191901$$
$$x_{30} = -36433.9712978862$$
$$x_{31} = -22024.5389490713$$
$$x_{32} = -27957.8546684026$$
$$x_{33} = 18659.5837299829$$
$$x_{34} = -24567.3936422505$$
$$x_{35} = 36459.5046194954$$
$$x_{36} = 33069.0627178099$$
$$x_{37} = 26288.1596384684$$
$$x_{38} = -26262.6255803404$$
$$x_{39} = -33891.1402146696$$
$$x_{40} = -31348.3061463201$$
$$x_{41} = -38976.79998085$$
$$x_{42} = -38129.1906568808$$
$$x_{43} = 31373.8397474042$$
$$x_{44} = 33916.6736603263$$
$$x_{45} = -21176.9184729939$$
$$x_{46} = 23745.3108174977$$
$$x_{47} = 30526.2276593423$$
$$x_{48} = 40697.5511560972$$
$$x_{49} = 39849.942279706$$
$$x_{50} = 20354.8317230213$$
$$x_{51} = 21202.4533555068$$
$$x_{52} = -25415.0100032502$$
$$x_{53} = 35611.894588966$$
$$x_{54} = -41519.6267050161$$
$$x_{55} = 42392.7683528465$$
$$x_{56} = 32221.4514230046$$
$$x_{57} = -40672.0179919024$$
$$x_{58} = -19481.6732724314$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 - \frac{2 \left(2 x - e^{\frac{1}{x}} + 2 + \frac{e^{\frac{1}{x}}}{x}\right)}{x} - \frac{2 \left(- x^{2} + x e^{\frac{1}{x}} - 2 x + 3\right)}{x^{2}} - \frac{e^{\frac{1}{x}}}{x^{3}}}{x}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{2 - \frac{2 \left(2 x - e^{\frac{1}{x}} + 2 + \frac{e^{\frac{1}{x}}}{x}\right)}{x} - \frac{2 \left(- x^{2} + x e^{\frac{1}{x}} - 2 x + 3\right)}{x^{2}} - \frac{e^{\frac{1}{x}}}{x^{3}}}{x}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- 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:
Concave at the intervals
$$\left[39849.942279706, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -42367.235238804\right]$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x}\right) = \infty$$
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 (x^2 + 2*x - 3 - x*E^(1/x))/x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x^{2}}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x^{2}}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x^{2}}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x^{2}}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
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:
$$\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x} = - \frac{x^{2} - 2 x + x e^{- \frac{1}{x}} - 3}{x}$$
- No
$$\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x} = \frac{x^{2} - 2 x + x e^{- \frac{1}{x}} - 3}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x} = - \frac{x^{2} - 2 x + x e^{- \frac{1}{x}} - 3}{x}$$
- No
$$\frac{- e^{\frac{1}{x}} x + \left(\left(x^{2} + 2 x\right) - 3\right)}{x} = \frac{x^{2} - 2 x + x e^{- \frac{1}{x}} - 3}{x}$$
- No
so, the function
not is
neither even, nor odd