Mister Exam
Graphing y = 1/(log(x)^3)-1
The solution
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1
f(x) = ------- - 1
3
log (x)
$$f{\left(x \right)} = -1 + \frac{1}{\log{\left(x \right)}^{3}}$$
f = -1 + 1/(log(x)^3)
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:
$$-1 + \frac{1}{\log{\left(x \right)}^{3}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = e$$
Numerical solution
$$x_{1} = 2.71828182845905$$
so we need to solve the equation:
$$-1 + \frac{1}{\log{\left(x \right)}^{3}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = e$$
Numerical solution
$$x_{1} = 2.71828182845905$$
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
$$- \frac{3}{x \log{\left(x \right)} \log{\left(x \right)}^{3}} = 0$$
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
$$- \frac{3}{x \log{\left(x \right)} \log{\left(x \right)}^{3}} = 0$$
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
$$\frac{3 \left(1 + \frac{4}{\log{\left(x \right)}}\right)}{x^{2} \log{\left(x \right)}^{4}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{-4}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{3 \left(1 + \frac{4}{\log{\left(x \right)}}\right)}{x^{2} \log{\left(x \right)}^{4}}\right) = -\infty$$
$$\lim_{x \to 1^+}\left(\frac{3 \left(1 + \frac{4}{\log{\left(x \right)}}\right)}{x^{2} \log{\left(x \right)}^{4}}\right) = \infty$$
- 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
$$\left(-\infty, e^{-4}\right]$$
Convex at the intervals
$$\left[e^{-4}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{3 \left(1 + \frac{4}{\log{\left(x \right)}}\right)}{x^{2} \log{\left(x \right)}^{4}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{-4}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{3 \left(1 + \frac{4}{\log{\left(x \right)}}\right)}{x^{2} \log{\left(x \right)}^{4}}\right) = -\infty$$
$$\lim_{x \to 1^+}\left(\frac{3 \left(1 + \frac{4}{\log{\left(x \right)}}\right)}{x^{2} \log{\left(x \right)}^{4}}\right) = \infty$$
- 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
$$\left(-\infty, e^{-4}\right]$$
Convex at the intervals
$$\left[e^{-4}, \infty\right)$$
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
$$\lim_{x \to -\infty}\left(-1 + \frac{1}{\log{\left(x \right)}^{3}}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(-1 + \frac{1}{\log{\left(x \right)}^{3}}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
$$\lim_{x \to -\infty}\left(-1 + \frac{1}{\log{\left(x \right)}^{3}}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(-1 + \frac{1}{\log{\left(x \right)}^{3}}\right) = -1$$
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 1/(log(x)^3) - 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-1 + \frac{1}{\log{\left(x \right)}^{3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{1}{\log{\left(x \right)}^{3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{-1 + \frac{1}{\log{\left(x \right)}^{3}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{1}{\log{\left(x \right)}^{3}}}{x}\right) = 0$$
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:
$$-1 + \frac{1}{\log{\left(x \right)}^{3}} = -1 + \frac{1}{\log{\left(- x \right)}^{3}}$$
- No
$$-1 + \frac{1}{\log{\left(x \right)}^{3}} = 1 - \frac{1}{\log{\left(- x \right)}^{3}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-1 + \frac{1}{\log{\left(x \right)}^{3}} = -1 + \frac{1}{\log{\left(- x \right)}^{3}}$$
- No
$$-1 + \frac{1}{\log{\left(x \right)}^{3}} = 1 - \frac{1}{\log{\left(- x \right)}^{3}}$$
- No
so, the function
not is
neither even, nor odd