Mister Exam
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-
How to use it?
- Graphing y =:
- x^3-6x
-
x^3+3x^2-9x+5
- x+2
- log(0.5,x)
-
Identical expressions
- log(zero . five ,x)
- logarithm of (0.5,x)
- logarithm of (zero . five ,x)
- log0.5,x
Graphing y = log(0.5,x)
The solution
You have entered
[src]
log(1/2)
f(x) = --------
log(x)
$$f{\left(x \right)} = \frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}}$$
f = log(1/2)/log(x)
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:
$$\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}} = 0$$
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:
$$\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}} = 0$$
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
$$- \frac{\log{\left(\frac{1}{2} \right)}}{x \log{\left(x \right)}^{2}} = 0$$
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
$$- \frac{\log{\left(\frac{1}{2} \right)}}{x \log{\left(x \right)}^{2}} = 0$$
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
$$\frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(\frac{1}{2} \right)}}{x^{2} \log{\left(x \right)}^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{-2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(\frac{1}{2} \right)}}{x^{2} \log{\left(x \right)}^{2}}\right) = \infty$$
$$\lim_{x \to 1^+}\left(\frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(\frac{1}{2} \right)}}{x^{2} \log{\left(x \right)}^{2}}\right) = -\infty$$
- 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
$$\left[e^{-2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, e^{-2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(\frac{1}{2} \right)}}{x^{2} \log{\left(x \right)}^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = e^{-2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(\frac{1}{2} \right)}}{x^{2} \log{\left(x \right)}^{2}}\right) = \infty$$
$$\lim_{x \to 1^+}\left(\frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(\frac{1}{2} \right)}}{x^{2} \log{\left(x \right)}^{2}}\right) = -\infty$$
- 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
$$\left[e^{-2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, e^{-2}\right]$$
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
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}}\right) = 0$$
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 log(1/2)/log(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{1}{2} \right)}}{x \log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{1}{2} \right)}}{x \log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{1}{2} \right)}}{x \log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{1}{2} \right)}}{x \log{\left(x \right)}}\right) = 0$$
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:
$$\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}} = \frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(- x \right)}}$$
- No
$$\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}} = - \frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}} = \frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(- x \right)}}$$
- No
$$\frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(x \right)}} = - \frac{\log{\left(\frac{1}{2} \right)}}{\log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd