Mister Exam
Graphing y = 2lnx/(x-2)-1
The solution
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2*log(x)
f(x) = -------- - 1
x - 2
$$f{\left(x \right)} = -1 + \frac{2 \log{\left(x \right)}}{x - 2}$$
f = -1 + (2*log(x))/(x - 2)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 2$$
$$x_{1} = 2$$
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:
$$-1 + \frac{2 \log{\left(x \right)}}{x - 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - 2 W\left(- \frac{1}{2 e}\right)$$
$$x_{2} = - 2 W_{-1}\left(- \frac{1}{2 e}\right)$$
Numerical solution
$$x_{1} = 0.463921905973069$$
$$x_{2} = 5.35669398003332$$
so we need to solve the equation:
$$-1 + \frac{2 \log{\left(x \right)}}{x - 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - 2 W\left(- \frac{1}{2 e}\right)$$
$$x_{2} = - 2 W_{-1}\left(- \frac{1}{2 e}\right)$$
Numerical solution
$$x_{1} = 0.463921905973069$$
$$x_{2} = 5.35669398003332$$
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
$$- \frac{2 \log{\left(x \right)}}{\left(x - 2\right)^{2}} + \frac{2}{x \left(x - 2\right)} = 0$$
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
$$- \frac{2 \log{\left(x \right)}}{\left(x - 2\right)^{2}} + \frac{2}{x \left(x - 2\right)} = 0$$
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
$$\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x - 2\right)^{2}} - \frac{2}{x \left(x - 2\right)} - \frac{1}{x^{2}}\right)}{x - 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 47764.2722655984$$
$$x_{2} = 51102.163330264$$
$$x_{3} = 38807.1423661755$$
$$x_{4} = 39931.8376131618$$
$$x_{5} = 46649.306437951$$
$$x_{6} = 43296.8041499171$$
$$x_{7} = 45533.1039315341$$
$$x_{8} = 56644.1060413689$$
$$x_{9} = 30883.9054427158$$
$$x_{10} = 57749.5929537082$$
$$x_{11} = 53321.9578958258$$
$$x_{12} = 49990.6618154252$$
$$x_{13} = 55537.6895261803$$
$$x_{14} = 44415.619329713$$
$$x_{15} = 42176.6065378353$$
$$x_{16} = 41054.9709207592$$
$$x_{17} = 54430.3163216977$$
$$x_{18} = 33157.5966083773$$
$$x_{19} = 48878.0440358842$$
$$x_{20} = 36552.7830628557$$
$$x_{21} = 37680.8158508515$$
$$x_{22} = 52212.5841572977$$
$$x_{23} = 32021.8485312361$$
$$x_{24} = 35422.9626334362$$
$$x_{25} = 34291.2660279171$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 2$$
$$\lim_{x \to 2^-}\left(\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x - 2\right)^{2}} - \frac{2}{x \left(x - 2\right)} - \frac{1}{x^{2}}\right)}{x - 2}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x - 2\right)^{2}} - \frac{2}{x \left(x - 2\right)} - \frac{1}{x^{2}}\right)}{x - 2}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 2$$
- 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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x - 2\right)^{2}} - \frac{2}{x \left(x - 2\right)} - \frac{1}{x^{2}}\right)}{x - 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 47764.2722655984$$
$$x_{2} = 51102.163330264$$
$$x_{3} = 38807.1423661755$$
$$x_{4} = 39931.8376131618$$
$$x_{5} = 46649.306437951$$
$$x_{6} = 43296.8041499171$$
$$x_{7} = 45533.1039315341$$
$$x_{8} = 56644.1060413689$$
$$x_{9} = 30883.9054427158$$
$$x_{10} = 57749.5929537082$$
$$x_{11} = 53321.9578958258$$
$$x_{12} = 49990.6618154252$$
$$x_{13} = 55537.6895261803$$
$$x_{14} = 44415.619329713$$
$$x_{15} = 42176.6065378353$$
$$x_{16} = 41054.9709207592$$
$$x_{17} = 54430.3163216977$$
$$x_{18} = 33157.5966083773$$
$$x_{19} = 48878.0440358842$$
$$x_{20} = 36552.7830628557$$
$$x_{21} = 37680.8158508515$$
$$x_{22} = 52212.5841572977$$
$$x_{23} = 32021.8485312361$$
$$x_{24} = 35422.9626334362$$
$$x_{25} = 34291.2660279171$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 2$$
$$\lim_{x \to 2^-}\left(\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x - 2\right)^{2}} - \frac{2}{x \left(x - 2\right)} - \frac{1}{x^{2}}\right)}{x - 2}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x - 2\right)^{2}} - \frac{2}{x \left(x - 2\right)} - \frac{1}{x^{2}}\right)}{x - 2}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 2$$
- 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = 2$$
$$x_{1} = 2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(-1 + \frac{2 \log{\left(x \right)}}{x - 2}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(-1 + \frac{2 \log{\left(x \right)}}{x - 2}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
$$\lim_{x \to -\infty}\left(-1 + \frac{2 \log{\left(x \right)}}{x - 2}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(-1 + \frac{2 \log{\left(x \right)}}{x - 2}\right) = -1$$
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 (2*log(x))/(x - 2) - 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-1 + \frac{2 \log{\left(x \right)}}{x - 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{2 \log{\left(x \right)}}{x - 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{-1 + \frac{2 \log{\left(x \right)}}{x - 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{2 \log{\left(x \right)}}{x - 2}}{x}\right) = 0$$
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:
$$-1 + \frac{2 \log{\left(x \right)}}{x - 2} = -1 + \frac{2 \log{\left(- x \right)}}{- x - 2}$$
- No
$$-1 + \frac{2 \log{\left(x \right)}}{x - 2} = 1 - \frac{2 \log{\left(- x \right)}}{- x - 2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-1 + \frac{2 \log{\left(x \right)}}{x - 2} = -1 + \frac{2 \log{\left(- x \right)}}{- x - 2}$$
- No
$$-1 + \frac{2 \log{\left(x \right)}}{x - 2} = 1 - \frac{2 \log{\left(- x \right)}}{- x - 2}$$
- No
so, the function
not is
neither even, nor odd