Mister Exam
Graphing y = (x+1)^(-1/(x-1))
The solution
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-1
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x - 1
f(x) = (x + 1)
$$f{\left(x \right)} = \left(x + 1\right)^{- \frac{1}{x - 1}}$$
f = (x + 1)^(-1/(x - 1))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1$$
$$x_{1} = 1$$
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:
$$\left(x + 1\right)^{- \frac{1}{x - 1}} = 0$$
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:
$$\left(x + 1\right)^{- \frac{1}{x - 1}} = 0$$
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
$$\left(x + 1\right)^{- \frac{1}{x - 1}} \left(- \frac{1}{\left(x - 1\right) \left(x + 1\right)} + \frac{\log{\left(x + 1 \right)}}{\left(x - 1\right)^{2}}\right) = 0$$
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
$$\left(x + 1\right)^{- \frac{1}{x - 1}} \left(- \frac{1}{\left(x - 1\right) \left(x + 1\right)} + \frac{\log{\left(x + 1 \right)}}{\left(x - 1\right)^{2}}\right) = 0$$
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
$$\frac{\left(x + 1\right)^{- \frac{1}{x - 1}} \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x - 1}\right)^{2}}{x - 1} + \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 34959.9177328973$$
$$x_{2} = 32674.0393112418$$
$$x_{3} = 58541.5194191283$$
$$x_{4} = 36099.3928798639$$
$$x_{5} = -0.449459108144713$$
$$x_{6} = 47391.1048906159$$
$$x_{7} = 56320.4735265899$$
$$x_{8} = 41766.5548123219$$
$$x_{9} = 29225.7459556974$$
$$x_{10} = 45145.8340124552$$
$$x_{11} = 52980.8904212903$$
$$x_{12} = 50748.7505171028$$
$$x_{13} = 40636.8355157506$$
$$x_{14} = 51865.4209527795$$
$$x_{15} = 37236.7193567847$$
$$x_{16} = 57431.5093027755$$
$$x_{17} = 26911.8192489273$$
$$x_{18} = 28070.4293666748$$
$$x_{19} = 38372.0044629403$$
$$x_{20} = 54095.1980555699$$
$$x_{21} = 49630.8375895814$$
$$x_{22} = 59650.5335724874$$
$$x_{23} = 25749.663458726$$
$$x_{24} = 48511.6380395588$$
$$x_{25} = 30377.9915735878$$
$$x_{26} = 39505.3464389136$$
$$x_{27} = 24583.6744114342$$
$$x_{28} = 42894.5811069$$
$$x_{29} = 55208.38078875$$
$$x_{30} = 31527.3638619932$$
$$x_{31} = 46269.18805139$$
$$x_{32} = 44020.9855038634$$
$$x_{33} = 33818.1763243429$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{\left(x + 1\right)^{- \frac{1}{x - 1}} \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x - 1}\right)^{2}}{x - 1} + \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1}\right) = \infty$$
$$\lim_{x \to 1^+}\left(\frac{\left(x + 1\right)^{- \frac{1}{x - 1}} \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x - 1}\right)^{2}}{x - 1} + \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1}\right) = 0$$
- the limits are not equal, so
$$x_{1} = 1$$
- 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
$$\left[-0.449459108144713, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -0.449459108144713\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(x + 1\right)^{- \frac{1}{x - 1}} \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x - 1}\right)^{2}}{x - 1} + \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 34959.9177328973$$
$$x_{2} = 32674.0393112418$$
$$x_{3} = 58541.5194191283$$
$$x_{4} = 36099.3928798639$$
$$x_{5} = -0.449459108144713$$
$$x_{6} = 47391.1048906159$$
$$x_{7} = 56320.4735265899$$
$$x_{8} = 41766.5548123219$$
$$x_{9} = 29225.7459556974$$
$$x_{10} = 45145.8340124552$$
$$x_{11} = 52980.8904212903$$
$$x_{12} = 50748.7505171028$$
$$x_{13} = 40636.8355157506$$
$$x_{14} = 51865.4209527795$$
$$x_{15} = 37236.7193567847$$
$$x_{16} = 57431.5093027755$$
$$x_{17} = 26911.8192489273$$
$$x_{18} = 28070.4293666748$$
$$x_{19} = 38372.0044629403$$
$$x_{20} = 54095.1980555699$$
$$x_{21} = 49630.8375895814$$
$$x_{22} = 59650.5335724874$$
$$x_{23} = 25749.663458726$$
$$x_{24} = 48511.6380395588$$
$$x_{25} = 30377.9915735878$$
$$x_{26} = 39505.3464389136$$
$$x_{27} = 24583.6744114342$$
$$x_{28} = 42894.5811069$$
$$x_{29} = 55208.38078875$$
$$x_{30} = 31527.3638619932$$
$$x_{31} = 46269.18805139$$
$$x_{32} = 44020.9855038634$$
$$x_{33} = 33818.1763243429$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{\left(x + 1\right)^{- \frac{1}{x - 1}} \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x - 1}\right)^{2}}{x - 1} + \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1}\right) = \infty$$
$$\lim_{x \to 1^+}\left(\frac{\left(x + 1\right)^{- \frac{1}{x - 1}} \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{\left(\frac{1}{x + 1} - \frac{\log{\left(x + 1 \right)}}{x - 1}\right)^{2}}{x - 1} + \frac{2}{\left(x - 1\right) \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2}}\right)}{x - 1}\right) = 0$$
- the limits are not equal, so
$$x_{1} = 1$$
- 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
$$\left[-0.449459108144713, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -0.449459108144713\right]$$
Vertical asymptotes
Have:
$$x_{1} = 1$$
$$x_{1} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left(x + 1\right)^{- \frac{1}{x - 1}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(x + 1\right)^{- \frac{1}{x - 1}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty} \left(x + 1\right)^{- \frac{1}{x - 1}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(x + 1\right)^{- \frac{1}{x - 1}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x + 1)^(-1/(x - 1)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)^{- \frac{1}{x - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{- \frac{1}{x - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)^{- \frac{1}{x - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{- \frac{1}{x - 1}}}{x}\right) = 0$$
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:
$$\left(x + 1\right)^{- \frac{1}{x - 1}} = \left(1 - x\right)^{- \frac{1}{- x - 1}}$$
- No
$$\left(x + 1\right)^{- \frac{1}{x - 1}} = - \left(1 - x\right)^{- \frac{1}{- x - 1}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(x + 1\right)^{- \frac{1}{x - 1}} = \left(1 - x\right)^{- \frac{1}{- x - 1}}$$
- No
$$\left(x + 1\right)^{- \frac{1}{x - 1}} = - \left(1 - x\right)^{- \frac{1}{- x - 1}}$$
- No
so, the function
not is
neither even, nor odd