Mister Exam
Graphing y = sign(x^2-4)x^2
The solution
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/ 2 \ 2 f(x) = sign\x - 4/*x
$$f{\left(x \right)} = x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}$$
f = x^2*sign(x^2 - 4)
The graph of the function
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
$$4 x^{3} \delta\left(x^{2} - 4\right) + 2 x \operatorname{sign}{\left(x^{2} - 4 \right)} = 0$$
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
$$4 x^{3} \delta\left(x^{2} - 4\right) + 2 x \operatorname{sign}{\left(x^{2} - 4 \right)} = 0$$
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
$$2 \left(2 x^{2} \left(2 x^{2} \delta^{\left( 1 \right)}\left( x^{2} - 4 \right) + \delta\left(x^{2} - 4\right)\right) + 8 x^{2} \delta\left(x^{2} - 4\right) + \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = 0$$
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
$$2 \left(2 x^{2} \left(2 x^{2} \delta^{\left( 1 \right)}\left( x^{2} - 4 \right) + \delta\left(x^{2} - 4\right)\right) + 8 x^{2} \delta\left(x^{2} - 4\right) + \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = 0$$
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
$$\lim_{x \to -\infty}\left(x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = \infty$$
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 sign(x^2 - 4)*x^2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(x \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \operatorname{sign}{\left(x^{2} - 4 \right)}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)} = x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}$$
- Yes
$$x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)} = - x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}$$
- No
so, the function
is
even
So, check:
$$x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)} = x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}$$
- Yes
$$x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)} = - x^{2} \operatorname{sign}{\left(x^{2} - 4 \right)}$$
- No
so, the function
is
even