Mister Exam
Graphing y = -(64x)/(x²-16)²
The solution
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-64*x
f(x) = ----------
2
/ 2 \
\x - 16/
$$f{\left(x \right)} = \frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}}$$
f = (-64*x)/(x^2 - 16)^2
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -4$$
$$x_{2} = 4$$
$$x_{1} = -4$$
$$x_{2} = 4$$
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:
$$\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 37301.7544113321$$
$$x_{2} = 39848.2926695335$$
$$x_{3} = -42262.9925603031$$
$$x_{4} = 0$$
$$x_{5} = 42394.3844016916$$
$$x_{6} = -39716.8799025336$$
$$x_{7} = -38868.0798379301$$
$$x_{8} = -38019.2265141401$$
$$x_{9} = 41545.7317146792$$
$$x_{10} = -41414.3333237521$$
$$x_{11} = -40565.6300435631$$
$$x_{12} = -37170.3162915843$$
$$x_{13} = 40697.0353978237$$
$$x_{14} = 38150.655617385$$
$$x_{15} = 38999.5005066015$$
so we need to solve the equation:
$$\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 37301.7544113321$$
$$x_{2} = 39848.2926695335$$
$$x_{3} = -42262.9925603031$$
$$x_{4} = 0$$
$$x_{5} = 42394.3844016916$$
$$x_{6} = -39716.8799025336$$
$$x_{7} = -38868.0798379301$$
$$x_{8} = -38019.2265141401$$
$$x_{9} = 41545.7317146792$$
$$x_{10} = -41414.3333237521$$
$$x_{11} = -40565.6300435631$$
$$x_{12} = -37170.3162915843$$
$$x_{13} = 40697.0353978237$$
$$x_{14} = 38150.655617385$$
$$x_{15} = 38999.5005066015$$
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
$$\frac{256 x^{2}}{\left(x^{2} - 16\right)^{3}} - \frac{64}{\left(x^{2} - 16\right)^{2}} = 0$$
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
$$\frac{256 x^{2}}{\left(x^{2} - 16\right)^{3}} - \frac{64}{\left(x^{2} - 16\right)^{2}} = 0$$
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
$$\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -4$$
$$x_{2} = 4$$
$$\lim_{x \to -4^-}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = \infty$$
$$\lim_{x \to -4^+}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = \infty$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 4^-}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = -\infty$$
$$\lim_{x \to 4^+}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = -\infty$$
- limits are equal, then skip the corresponding 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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -4$$
$$x_{2} = 4$$
$$\lim_{x \to -4^-}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = \infty$$
$$\lim_{x \to -4^+}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = \infty$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 4^-}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = -\infty$$
$$\lim_{x \to 4^+}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = -\infty$$
- limits are equal, then skip the corresponding 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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = -4$$
$$x_{2} = 4$$
$$x_{1} = -4$$
$$x_{2} = 4$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}}\right) = 0$$
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 (-64*x)/(x^2 - 16)^2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(- \frac{64}{\left(x^{2} - 16\right)^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(- \frac{64}{\left(x^{2} - 16\right)^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(- \frac{64}{\left(x^{2} - 16\right)^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(- \frac{64}{\left(x^{2} - 16\right)^{2}}\right) = 0$$
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:
$$\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = \frac{64 x}{\left(x^{2} - 16\right)^{2}}$$
- No
$$\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = - \frac{64 x}{\left(x^{2} - 16\right)^{2}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = \frac{64 x}{\left(x^{2} - 16\right)^{2}}$$
- No
$$\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = - \frac{64 x}{\left(x^{2} - 16\right)^{2}}$$
- No
so, the function
not is
neither even, nor odd