Mister Exam
Graphing y = atan(5^x+1)^3
The solution
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3/ x \ f(x) = atan \5 + 1/
$$f{\left(x \right)} = \operatorname{atan}^{3}{\left(5^{x} + 1 \right)}$$
f = atan(5^x + 1)^3
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:
$$\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = 0$$
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:
$$\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = 0$$
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
$$\frac{3 \cdot 5^{x} \log{\left(5 \right)} \operatorname{atan}^{2}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 1} = 0$$
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
$$\frac{3 \cdot 5^{x} \log{\left(5 \right)} \operatorname{atan}^{2}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 1} = 0$$
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
$$\frac{3 \cdot 5^{x} \left(- \frac{2 \cdot 5^{x} \left(5^{x} + 1\right) \operatorname{atan}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 1} + \frac{2 \cdot 5^{x}}{\left(5^{x} + 1\right)^{2} + 1} + \operatorname{atan}{\left(5^{x} + 1 \right)}\right) \log{\left(5 \right)}^{2} \operatorname{atan}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 95.2242768391714$$
$$x_{2} = 85.2242768391714$$
$$x_{3} = 21.2242894893908$$
$$x_{4} = -44.9148275920459$$
$$x_{5} = -60.9148275920459$$
$$x_{6} = -36.9148275920459$$
$$x_{7} = -66.9148275920459$$
$$x_{8} = -28.9148275920415$$
$$x_{9} = 93.2242768391714$$
$$x_{10} = -110.914827592046$$
$$x_{11} = -46.9148275920459$$
$$x_{12} = 43.2242768391714$$
$$x_{13} = -84.9148275920459$$
$$x_{14} = -90.9148275920459$$
$$x_{15} = -102.914827592046$$
$$x_{16} = 69.2242768391714$$
$$x_{17} = -112.914827592046$$
$$x_{18} = 31.2242768391727$$
$$x_{19} = -20.9148258621738$$
$$x_{20} = -82.9148275920459$$
$$x_{21} = -56.9148275920459$$
$$x_{22} = 39.2242768391714$$
$$x_{23} = 87.2242768391714$$
$$x_{24} = -70.9148275920459$$
$$x_{25} = -76.9148275920459$$
$$x_{26} = 53.2242768391714$$
$$x_{27} = 71.2242768391714$$
$$x_{28} = 23.2242773451776$$
$$x_{29} = 103.224276839171$$
$$x_{30} = -104.914827592046$$
$$x_{31} = 81.2242768391714$$
$$x_{32} = -68.9148275920459$$
$$x_{33} = 19.2245931342562$$
$$x_{34} = 0.542176246771954$$
$$x_{35} = 79.2242768391714$$
$$x_{36} = -38.9148275920459$$
$$x_{37} = 27.224276839981$$
$$x_{38} = 55.2242768391714$$
$$x_{39} = -26.9148275919352$$
$$x_{40} = -64.9148275920459$$
$$x_{41} = -50.9148275920459$$
$$x_{42} = -42.9148275920459$$
$$x_{43} = 41.2242768391714$$
$$x_{44} = 67.2242768391714$$
$$x_{45} = 63.2242768391714$$
$$x_{46} = -54.9148275920459$$
$$x_{47} = 45.2242768391714$$
$$x_{48} = 75.2242768391714$$
$$x_{49} = -92.9148275920459$$
$$x_{50} = -100.914827592046$$
$$x_{51} = 97.2242768391714$$
$$x_{52} = 107.224276839171$$
$$x_{53} = 101.224276839171$$
$$x_{54} = -52.9148275920459$$
$$x_{55} = -32.9148275920459$$
$$x_{56} = -74.9148275920459$$
$$x_{57} = 105.224276839171$$
$$x_{58} = 83.2242768391714$$
$$x_{59} = -80.9148275920459$$
$$x_{60} = 91.2242768391714$$
$$x_{61} = 51.2242768391714$$
$$x_{62} = -96.9148275920459$$
$$x_{63} = 77.2242768391714$$
$$x_{64} = -40.9148275920459$$
$$x_{65} = -72.9148275920459$$
$$x_{66} = -24.914827589278$$
$$x_{67} = 61.2242768391714$$
$$x_{68} = -78.9148275920459$$
$$x_{69} = 37.2242768391714$$
$$x_{70} = 113.224276839171$$
$$x_{71} = -94.9148275920459$$
$$x_{72} = 109.224276839171$$
$$x_{73} = 47.2242768391714$$
$$x_{74} = 111.224276839171$$
$$x_{75} = 57.2242768391714$$
$$x_{76} = -108.914827592046$$
$$x_{77} = -22.9148275228492$$
$$x_{78} = 25.2242768594116$$
$$x_{79} = 89.2242768391714$$
$$x_{80} = 29.2242768392038$$
$$x_{81} = 59.2242768391714$$
$$x_{82} = 35.2242768391714$$
$$x_{83} = 99.2242768391714$$
$$x_{84} = -86.9148275920459$$
$$x_{85} = -106.914827592046$$
$$x_{86} = 49.2242768391714$$
$$x_{87} = -98.9148275920459$$
$$x_{88} = -62.9148275920459$$
$$x_{89} = 33.2242768391715$$
$$x_{90} = 65.2242768391714$$
$$x_{91} = -88.9148275920459$$
$$x_{92} = -48.9148275920459$$
$$x_{93} = -58.9148275920459$$
$$x_{94} = 73.2242768391714$$
$$x_{95} = -18.914784374042$$
$$x_{96} = -30.9148275920457$$
$$x_{97} = -34.9148275920459$$
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
$$\left(-\infty, 0.542176246771954\right]$$
Convex at the intervals
$$\left[0.542176246771954, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{3 \cdot 5^{x} \left(- \frac{2 \cdot 5^{x} \left(5^{x} + 1\right) \operatorname{atan}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 1} + \frac{2 \cdot 5^{x}}{\left(5^{x} + 1\right)^{2} + 1} + \operatorname{atan}{\left(5^{x} + 1 \right)}\right) \log{\left(5 \right)}^{2} \operatorname{atan}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 95.2242768391714$$
$$x_{2} = 85.2242768391714$$
$$x_{3} = 21.2242894893908$$
$$x_{4} = -44.9148275920459$$
$$x_{5} = -60.9148275920459$$
$$x_{6} = -36.9148275920459$$
$$x_{7} = -66.9148275920459$$
$$x_{8} = -28.9148275920415$$
$$x_{9} = 93.2242768391714$$
$$x_{10} = -110.914827592046$$
$$x_{11} = -46.9148275920459$$
$$x_{12} = 43.2242768391714$$
$$x_{13} = -84.9148275920459$$
$$x_{14} = -90.9148275920459$$
$$x_{15} = -102.914827592046$$
$$x_{16} = 69.2242768391714$$
$$x_{17} = -112.914827592046$$
$$x_{18} = 31.2242768391727$$
$$x_{19} = -20.9148258621738$$
$$x_{20} = -82.9148275920459$$
$$x_{21} = -56.9148275920459$$
$$x_{22} = 39.2242768391714$$
$$x_{23} = 87.2242768391714$$
$$x_{24} = -70.9148275920459$$
$$x_{25} = -76.9148275920459$$
$$x_{26} = 53.2242768391714$$
$$x_{27} = 71.2242768391714$$
$$x_{28} = 23.2242773451776$$
$$x_{29} = 103.224276839171$$
$$x_{30} = -104.914827592046$$
$$x_{31} = 81.2242768391714$$
$$x_{32} = -68.9148275920459$$
$$x_{33} = 19.2245931342562$$
$$x_{34} = 0.542176246771954$$
$$x_{35} = 79.2242768391714$$
$$x_{36} = -38.9148275920459$$
$$x_{37} = 27.224276839981$$
$$x_{38} = 55.2242768391714$$
$$x_{39} = -26.9148275919352$$
$$x_{40} = -64.9148275920459$$
$$x_{41} = -50.9148275920459$$
$$x_{42} = -42.9148275920459$$
$$x_{43} = 41.2242768391714$$
$$x_{44} = 67.2242768391714$$
$$x_{45} = 63.2242768391714$$
$$x_{46} = -54.9148275920459$$
$$x_{47} = 45.2242768391714$$
$$x_{48} = 75.2242768391714$$
$$x_{49} = -92.9148275920459$$
$$x_{50} = -100.914827592046$$
$$x_{51} = 97.2242768391714$$
$$x_{52} = 107.224276839171$$
$$x_{53} = 101.224276839171$$
$$x_{54} = -52.9148275920459$$
$$x_{55} = -32.9148275920459$$
$$x_{56} = -74.9148275920459$$
$$x_{57} = 105.224276839171$$
$$x_{58} = 83.2242768391714$$
$$x_{59} = -80.9148275920459$$
$$x_{60} = 91.2242768391714$$
$$x_{61} = 51.2242768391714$$
$$x_{62} = -96.9148275920459$$
$$x_{63} = 77.2242768391714$$
$$x_{64} = -40.9148275920459$$
$$x_{65} = -72.9148275920459$$
$$x_{66} = -24.914827589278$$
$$x_{67} = 61.2242768391714$$
$$x_{68} = -78.9148275920459$$
$$x_{69} = 37.2242768391714$$
$$x_{70} = 113.224276839171$$
$$x_{71} = -94.9148275920459$$
$$x_{72} = 109.224276839171$$
$$x_{73} = 47.2242768391714$$
$$x_{74} = 111.224276839171$$
$$x_{75} = 57.2242768391714$$
$$x_{76} = -108.914827592046$$
$$x_{77} = -22.9148275228492$$
$$x_{78} = 25.2242768594116$$
$$x_{79} = 89.2242768391714$$
$$x_{80} = 29.2242768392038$$
$$x_{81} = 59.2242768391714$$
$$x_{82} = 35.2242768391714$$
$$x_{83} = 99.2242768391714$$
$$x_{84} = -86.9148275920459$$
$$x_{85} = -106.914827592046$$
$$x_{86} = 49.2242768391714$$
$$x_{87} = -98.9148275920459$$
$$x_{88} = -62.9148275920459$$
$$x_{89} = 33.2242768391715$$
$$x_{90} = 65.2242768391714$$
$$x_{91} = -88.9148275920459$$
$$x_{92} = -48.9148275920459$$
$$x_{93} = -58.9148275920459$$
$$x_{94} = 73.2242768391714$$
$$x_{95} = -18.914784374042$$
$$x_{96} = -30.9148275920457$$
$$x_{97} = -34.9148275920459$$
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
$$\left(-\infty, 0.542176246771954\right]$$
Convex at the intervals
$$\left[0.542176246771954, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \frac{\pi^{3}}{64}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\pi^{3}}{64}$$
$$\lim_{x \to \infty} \operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \frac{\pi^{3}}{8}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\pi^{3}}{8}$$
$$\lim_{x \to -\infty} \operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \frac{\pi^{3}}{64}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\pi^{3}}{64}$$
$$\lim_{x \to \infty} \operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \frac{\pi^{3}}{8}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\pi^{3}}{8}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(5^x + 1)^3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{3}{\left(5^{x} + 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{3}{\left(5^{x} + 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{3}{\left(5^{x} + 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{3}{\left(5^{x} + 1 \right)}}{x}\right) = 0$$
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:
$$\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \operatorname{atan}^{3}{\left(1 + 5^{- x} \right)}$$
- No
$$\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = - \operatorname{atan}^{3}{\left(1 + 5^{- x} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \operatorname{atan}^{3}{\left(1 + 5^{- x} \right)}$$
- No
$$\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = - \operatorname{atan}^{3}{\left(1 + 5^{- x} \right)}$$
- No
so, the function
not is
neither even, nor odd