Mister Exam
Graphing y = (-1+x*(-1-5*x))*exp(-4*x)
The solution
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-4*x f(x) = (-1 + x*(-1 - 5*x))*e
$$f{\left(x \right)} = \left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}$$
f = (x*(-5*x - 1) - 1)*exp(-4*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:
$$\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 87.4598473975713$$
$$x_{2} = 71.4671329611604$$
$$x_{3} = 95.4571523435427$$
$$x_{4} = 67.4695213707883$$
$$x_{5} = 85.4606030765202$$
$$x_{6} = 51.4830361975311$$
$$x_{7} = 105.454376550757$$
$$x_{8} = 41.4972282607378$$
$$x_{9} = 43.4938252209702$$
$$x_{10} = 89.4591267644358$$
$$x_{11} = 37.5052281025737$$
$$x_{12} = 83.4613964218887$$
$$x_{13} = 49.4853833181854$$
$$x_{14} = 103.454887409586$$
$$x_{15} = 75.4650090390186$$
$$x_{16} = 63.4722269658107$$
$$x_{17} = 47.4879427375979$$
$$x_{18} = 17.6130141728092$$
$$x_{19} = 11.7497125515628$$
$$x_{20} = 93.4577813149409$$
$$x_{21} = 27.5368015935747$$
$$x_{22} = 29.5285572612671$$
$$x_{23} = 101.455419065557$$
$$x_{24} = 15.6433868867448$$
$$x_{25} = 81.4622303222827$$
$$x_{26} = 39.5010072252958$$
$$x_{27} = 79.4631079694713$$
$$x_{28} = 55.4788812527068$$
$$x_{29} = 21.5724341623746$$
$$x_{30} = 59.4753174443243$$
$$x_{31} = 9.85707471559485$$
$$x_{32} = 25.5465338221062$$
$$x_{33} = 99.4559728151677$$
$$x_{34} = 35.5099733005247$$
$$x_{35} = 53.4808760032941$$
$$x_{36} = 31.5214834344543$$
$$x_{37} = 19.5902031682541$$
$$x_{38} = 33.5153470994371$$
$$x_{39} = 23.558197821335$$
$$x_{40} = 61.4737191365914$$
$$x_{41} = 65.4708306970877$$
$$x_{42} = 69.4682911098761$$
$$x_{43} = 45.4907446402201$$
$$x_{44} = 77.4640328992377$$
$$x_{45} = 13.6858785168085$$
$$x_{46} = 73.4660407636888$$
$$x_{47} = 97.456550064977$$
$$x_{48} = 57.4770336345057$$
$$x_{49} = 91.4584387941273$$
so we need to solve the equation:
$$\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 87.4598473975713$$
$$x_{2} = 71.4671329611604$$
$$x_{3} = 95.4571523435427$$
$$x_{4} = 67.4695213707883$$
$$x_{5} = 85.4606030765202$$
$$x_{6} = 51.4830361975311$$
$$x_{7} = 105.454376550757$$
$$x_{8} = 41.4972282607378$$
$$x_{9} = 43.4938252209702$$
$$x_{10} = 89.4591267644358$$
$$x_{11} = 37.5052281025737$$
$$x_{12} = 83.4613964218887$$
$$x_{13} = 49.4853833181854$$
$$x_{14} = 103.454887409586$$
$$x_{15} = 75.4650090390186$$
$$x_{16} = 63.4722269658107$$
$$x_{17} = 47.4879427375979$$
$$x_{18} = 17.6130141728092$$
$$x_{19} = 11.7497125515628$$
$$x_{20} = 93.4577813149409$$
$$x_{21} = 27.5368015935747$$
$$x_{22} = 29.5285572612671$$
$$x_{23} = 101.455419065557$$
$$x_{24} = 15.6433868867448$$
$$x_{25} = 81.4622303222827$$
$$x_{26} = 39.5010072252958$$
$$x_{27} = 79.4631079694713$$
$$x_{28} = 55.4788812527068$$
$$x_{29} = 21.5724341623746$$
$$x_{30} = 59.4753174443243$$
$$x_{31} = 9.85707471559485$$
$$x_{32} = 25.5465338221062$$
$$x_{33} = 99.4559728151677$$
$$x_{34} = 35.5099733005247$$
$$x_{35} = 53.4808760032941$$
$$x_{36} = 31.5214834344543$$
$$x_{37} = 19.5902031682541$$
$$x_{38} = 33.5153470994371$$
$$x_{39} = 23.558197821335$$
$$x_{40} = 61.4737191365914$$
$$x_{41} = 65.4708306970877$$
$$x_{42} = 69.4682911098761$$
$$x_{43} = 45.4907446402201$$
$$x_{44} = 77.4640328992377$$
$$x_{45} = 13.6858785168085$$
$$x_{46} = 73.4660407636888$$
$$x_{47} = 97.456550064977$$
$$x_{48} = 57.4770336345057$$
$$x_{49} = 91.4584387941273$$
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
$$\left(- 10 x - 1\right) e^{- 4 x} - 4 \left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x} = 0$$
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
$$\left(- 10 x - 1\right) e^{- 4 x} - 4 \left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x} = 0$$
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
$$2 \left(- 8 x \left(5 x + 1\right) + 40 x - 9\right) e^{- 4 x} = 0$$
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
$$2 \left(- 8 x \left(5 x + 1\right) + 40 x - 9\right) e^{- 4 x} = 0$$
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
$$\lim_{x \to -\infty}\left(\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}\right) = 0$$
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 (-1 + x*(-1 - 5*x))*exp(-4*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x}}{x}\right) = 0$$
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:
$$\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x} = \left(- x \left(5 x - 1\right) - 1\right) e^{4 x}$$
- No
$$\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x} = - \left(- x \left(5 x - 1\right) - 1\right) e^{4 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x} = \left(- x \left(5 x - 1\right) - 1\right) e^{4 x}$$
- No
$$\left(x \left(- 5 x - 1\right) - 1\right) e^{- 4 x} = - \left(- x \left(5 x - 1\right) - 1\right) e^{4 x}$$
- No
so, the function
not is
neither even, nor odd