Mister Exam
Graphing y = 2-arccot(x+3)
The solution
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f(x) = 2 - acot(x + 3)
$$f{\left(x \right)} = 2 - \operatorname{acot}{\left(x + 3 \right)}$$
f = 2 - acot(x + 3)
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:
$$2 - \operatorname{acot}{\left(x + 3 \right)} = 0$$
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:
$$2 - \operatorname{acot}{\left(x + 3 \right)} = 0$$
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
$$\frac{1}{\left(x + 3\right)^{2} + 1} = 0$$
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
$$\frac{1}{\left(x + 3\right)^{2} + 1} = 0$$
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
$$- \frac{2 \left(x + 3\right)}{\left(\left(x + 3\right)^{2} + 1\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -3$$
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
$$\left(-\infty, -3\right]$$
Convex at the intervals
$$\left[-3, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \left(x + 3\right)}{\left(\left(x + 3\right)^{2} + 1\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -3$$
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
$$\left(-\infty, -3\right]$$
Convex at the intervals
$$\left[-3, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(2 - \operatorname{acot}{\left(x + 3 \right)}\right) = 2 - \pi$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 2 - \pi$$
$$\lim_{x \to \infty}\left(2 - \operatorname{acot}{\left(x + 3 \right)}\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 2$$
$$\lim_{x \to -\infty}\left(2 - \operatorname{acot}{\left(x + 3 \right)}\right) = 2 - \pi$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 2 - \pi$$
$$\lim_{x \to \infty}\left(2 - \operatorname{acot}{\left(x + 3 \right)}\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 2$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2 - acot(x + 3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 - \operatorname{acot}{\left(x + 3 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 - \operatorname{acot}{\left(x + 3 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{2 - \operatorname{acot}{\left(x + 3 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 - \operatorname{acot}{\left(x + 3 \right)}}{x}\right) = 0$$
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:
$$2 - \operatorname{acot}{\left(x + 3 \right)} = \operatorname{acot}{\left(x - 3 \right)} + 2$$
- No
$$2 - \operatorname{acot}{\left(x + 3 \right)} = - \operatorname{acot}{\left(x - 3 \right)} - 2$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2 - \operatorname{acot}{\left(x + 3 \right)} = \operatorname{acot}{\left(x - 3 \right)} + 2$$
- No
$$2 - \operatorname{acot}{\left(x + 3 \right)} = - \operatorname{acot}{\left(x - 3 \right)} - 2$$
- No
so, the function
not is
neither even, nor odd