Mister Exam
Graphing y = x+arcctg(x/2)
The solution
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/x\
f(x) = x + acot|-|
\2/
$$f{\left(x \right)} = x + \operatorname{acot}{\left(\frac{x}{2} \right)}$$
f = x + acot(x/2)
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:
$$x + \operatorname{acot}{\left(\frac{x}{2} \right)} = 0$$
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:
$$x + \operatorname{acot}{\left(\frac{x}{2} \right)} = 0$$
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
$$1 - \frac{1}{2 \left(\frac{x^{2}}{4} + 1\right)} = 0$$
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
$$1 - \frac{1}{2 \left(\frac{x^{2}}{4} + 1\right)} = 0$$
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
$$\frac{x}{4 \left(\frac{x^{2}}{4} + 1\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{x}{4 \left(\frac{x^{2}}{4} + 1\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(x + \operatorname{acot}{\left(\frac{x}{2} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x + \operatorname{acot}{\left(\frac{x}{2} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x + \operatorname{acot}{\left(\frac{x}{2} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x + \operatorname{acot}{\left(\frac{x}{2} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x + acot(x/2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x + \operatorname{acot}{\left(\frac{x}{2} \right)}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{x + \operatorname{acot}{\left(\frac{x}{2} \right)}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{x + \operatorname{acot}{\left(\frac{x}{2} \right)}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{x + \operatorname{acot}{\left(\frac{x}{2} \right)}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
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:
$$x + \operatorname{acot}{\left(\frac{x}{2} \right)} = - x - \operatorname{acot}{\left(\frac{x}{2} \right)}$$
- No
$$x + \operatorname{acot}{\left(\frac{x}{2} \right)} = x + \operatorname{acot}{\left(\frac{x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x + \operatorname{acot}{\left(\frac{x}{2} \right)} = - x - \operatorname{acot}{\left(\frac{x}{2} \right)}$$
- No
$$x + \operatorname{acot}{\left(\frac{x}{2} \right)} = x + \operatorname{acot}{\left(\frac{x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd