Mister Exam
Graphing y = sign(x-x^3)
The solution
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/ 3\ f(x) = sign\x - x /
$$f{\left(x \right)} = \operatorname{sign}{\left(- x^{3} + x \right)}$$
f = sign(-x^3 + x)
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:
$$\operatorname{sign}{\left(- x^{3} + x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\operatorname{sign}{\left(- x^{3} + x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
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
$$- 2 \left(6 x \delta\left(x \left(x^{2} - 1\right)\right) + \left(3 x^{2} - 1\right)^{2} \delta^{\left( 1 \right)}\left( x \left(x^{2} - 1\right) \right)\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \left(6 x \delta\left(x \left(x^{2} - 1\right)\right) + \left(3 x^{2} - 1\right)^{2} \delta^{\left( 1 \right)}\left( x \left(x^{2} - 1\right) \right)\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{sign}{\left(- x^{3} + x \right)} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \operatorname{sign}{\left(- x^{3} + x \right)} = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
$$\lim_{x \to -\infty} \operatorname{sign}{\left(- x^{3} + x \right)} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \operatorname{sign}{\left(- x^{3} + x \right)} = -1$$
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 sign(x - x^3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{sign}{\left(- x^{3} + x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{sign}{\left(- x^{3} + x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{sign}{\left(- x^{3} + x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{sign}{\left(- x^{3} + x \right)}}{x}\right) = 0$$
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:
$$\operatorname{sign}{\left(- x^{3} + x \right)} = \operatorname{sign}{\left(x^{3} - x \right)}$$
- No
$$\operatorname{sign}{\left(- x^{3} + x \right)} = - \operatorname{sign}{\left(x^{3} - x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{sign}{\left(- x^{3} + x \right)} = \operatorname{sign}{\left(x^{3} - x \right)}$$
- No
$$\operatorname{sign}{\left(- x^{3} + x \right)} = - \operatorname{sign}{\left(x^{3} - x \right)}$$
- No
so, the function
not is
neither even, nor odd