Mister Exam
Graphing y = (1+(1+x)*exp(x))*exp(x)
The solution
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[src]
/ x\ x f(x) = \1 + (1 + x)*e /*e
$$f{\left(x \right)} = \left(\left(x + 1\right) e^{x} + 1\right) e^{x}$$
f = ((x + 1)*exp(x) + 1)*exp(x)
The graph of the function
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:
$$\left(\left(x + 1\right) e^{x} + 1\right) e^{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -102.872003083$$
$$x_{2} = -64.8720030830002$$
$$x_{3} = -36.8720060457829$$
$$x_{4} = -44.8720030845445$$
$$x_{5} = -92.8720030830002$$
$$x_{6} = -78.8720030830002$$
$$x_{7} = -106.872003083$$
$$x_{8} = -62.8720030830002$$
$$x_{9} = -110.872003083$$
$$x_{10} = -34.8720219445663$$
$$x_{11} = -48.8720030830335$$
$$x_{12} = -116.872003083$$
$$x_{13} = -104.872003083$$
$$x_{14} = -52.8720030830009$$
$$x_{15} = -58.8720030830002$$
$$x_{16} = -120.872003083$$
$$x_{17} = -118.872003083$$
$$x_{18} = -76.8720030830002$$
$$x_{19} = -32.8721200643545$$
$$x_{20} = -28.8759536128387$$
$$x_{21} = -46.8720030832278$$
$$x_{22} = -68.8720030830002$$
$$x_{23} = -72.8720030830002$$
$$x_{24} = -84.8720030830002$$
$$x_{25} = -56.8720030830002$$
$$x_{26} = -30.872702226286$$
$$x_{27} = -60.8720030830002$$
$$x_{28} = -100.872003083$$
$$x_{29} = -98.8720030830002$$
$$x_{30} = -94.8720030830002$$
$$x_{31} = -74.8720030830002$$
$$x_{32} = -50.872003083005$$
$$x_{33} = -88.8720030830002$$
$$x_{34} = -112.872003083$$
$$x_{35} = -108.872003083$$
$$x_{36} = -40.8720031522905$$
$$x_{37} = -54.8720030830003$$
$$x_{38} = -38.8720035394793$$
$$x_{39} = -114.872003083$$
$$x_{40} = -80.8720030830002$$
$$x_{41} = -42.8720030933943$$
$$x_{42} = -70.8720030830002$$
$$x_{43} = -66.8720030830002$$
$$x_{44} = -86.8720030830002$$
$$x_{45} = -90.8720030830002$$
$$x_{46} = -82.8720030830002$$
$$x_{47} = -96.8720030830002$$
so we need to solve the equation:
$$\left(\left(x + 1\right) e^{x} + 1\right) e^{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -102.872003083$$
$$x_{2} = -64.8720030830002$$
$$x_{3} = -36.8720060457829$$
$$x_{4} = -44.8720030845445$$
$$x_{5} = -92.8720030830002$$
$$x_{6} = -78.8720030830002$$
$$x_{7} = -106.872003083$$
$$x_{8} = -62.8720030830002$$
$$x_{9} = -110.872003083$$
$$x_{10} = -34.8720219445663$$
$$x_{11} = -48.8720030830335$$
$$x_{12} = -116.872003083$$
$$x_{13} = -104.872003083$$
$$x_{14} = -52.8720030830009$$
$$x_{15} = -58.8720030830002$$
$$x_{16} = -120.872003083$$
$$x_{17} = -118.872003083$$
$$x_{18} = -76.8720030830002$$
$$x_{19} = -32.8721200643545$$
$$x_{20} = -28.8759536128387$$
$$x_{21} = -46.8720030832278$$
$$x_{22} = -68.8720030830002$$
$$x_{23} = -72.8720030830002$$
$$x_{24} = -84.8720030830002$$
$$x_{25} = -56.8720030830002$$
$$x_{26} = -30.872702226286$$
$$x_{27} = -60.8720030830002$$
$$x_{28} = -100.872003083$$
$$x_{29} = -98.8720030830002$$
$$x_{30} = -94.8720030830002$$
$$x_{31} = -74.8720030830002$$
$$x_{32} = -50.872003083005$$
$$x_{33} = -88.8720030830002$$
$$x_{34} = -112.872003083$$
$$x_{35} = -108.872003083$$
$$x_{36} = -40.8720031522905$$
$$x_{37} = -54.8720030830003$$
$$x_{38} = -38.8720035394793$$
$$x_{39} = -114.872003083$$
$$x_{40} = -80.8720030830002$$
$$x_{41} = -42.8720030933943$$
$$x_{42} = -70.8720030830002$$
$$x_{43} = -66.8720030830002$$
$$x_{44} = -86.8720030830002$$
$$x_{45} = -90.8720030830002$$
$$x_{46} = -82.8720030830002$$
$$x_{47} = -96.8720030830002$$
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
$$\left(\left(x + 1\right) e^{x} + 1\right) e^{x} + \left(\left(x + 1\right) e^{x} + e^{x}\right) e^{x} = 0$$
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
$$\left(\left(x + 1\right) e^{x} + 1\right) e^{x} + \left(\left(x + 1\right) e^{x} + e^{x}\right) e^{x} = 0$$
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
$$\left(\left(x + 1\right) e^{x} + 2 \left(x + 2\right) e^{x} + \left(x + 3\right) e^{x} + 1\right) e^{x} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\left(\left(x + 1\right) e^{x} + 2 \left(x + 2\right) e^{x} + \left(x + 3\right) e^{x} + 1\right) e^{x} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\left(\left(x + 1\right) e^{x} + 1\right) e^{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\left(\left(x + 1\right) e^{x} + 1\right) e^{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\left(\left(x + 1\right) e^{x} + 1\right) e^{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\left(\left(x + 1\right) e^{x} + 1\right) e^{x}\right) = \infty$$
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 (1 + (1 + x)*exp(x))*exp(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\left(x + 1\right) e^{x} + 1\right) e^{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(\left(x + 1\right) e^{x} + 1\right) e^{x}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left(\left(x + 1\right) e^{x} + 1\right) e^{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(\left(x + 1\right) e^{x} + 1\right) e^{x}}{x}\right) = \infty$$
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:
$$\left(\left(x + 1\right) e^{x} + 1\right) e^{x} = \left(\left(1 - x\right) e^{- x} + 1\right) e^{- x}$$
- No
$$\left(\left(x + 1\right) e^{x} + 1\right) e^{x} = - \left(\left(1 - x\right) e^{- x} + 1\right) e^{- x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(\left(x + 1\right) e^{x} + 1\right) e^{x} = \left(\left(1 - x\right) e^{- x} + 1\right) e^{- x}$$
- No
$$\left(\left(x + 1\right) e^{x} + 1\right) e^{x} = - \left(\left(1 - x\right) e^{- x} + 1\right) e^{- x}$$
- No
so, the function
not is
neither even, nor odd