Mister Exam
Graphing y = (x^2+3*x)^(1/3)*cos((pi*x)/2)
The solution
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3 / 2 /pi*x\
f(x) = \/ x + 3*x *cos|----|
\ 2 /
$$f{\left(x \right)} = \sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}$$
f = (x^2 + 3*x)^(1/3)*cos((pi*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:
$$\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3$$
$$x_{2} = 0$$
$$x_{3} = 1$$
$$x_{4} = 3$$
Numerical solution
$$x_{1} = -29$$
$$x_{2} = 1$$
$$x_{3} = -33$$
$$x_{4} = -61$$
$$x_{5} = -59$$
$$x_{6} = -19$$
$$x_{7} = 71$$
$$x_{8} = -23$$
$$x_{9} = -1$$
$$x_{10} = -41$$
$$x_{11} = -2.99999999999964$$
$$x_{12} = 51$$
$$x_{13} = -2.99999999966621$$
$$x_{14} = -55$$
$$x_{15} = -35$$
$$x_{16} = -167$$
$$x_{17} = -31$$
$$x_{18} = -69$$
$$x_{19} = -25$$
$$x_{20} = -89$$
$$x_{21} = 399$$
$$x_{22} = -2.99999999998298$$
$$x_{23} = -27$$
$$x_{24} = 45$$
$$x_{25} = -67$$
$$x_{26} = -73$$
$$x_{27} = -47$$
$$x_{28} = -65$$
$$x_{29} = 59$$
$$x_{30} = -15$$
$$x_{31} = 69$$
$$x_{32} = -63$$
$$x_{33} = -75$$
$$x_{34} = -39$$
$$x_{35} = 93$$
$$x_{36} = -79$$
$$x_{37} = 85$$
$$x_{38} = -5$$
$$x_{39} = 57$$
$$x_{40} = 79$$
$$x_{41} = -53$$
$$x_{42} = -9$$
$$x_{43} = -2.999999999994$$
$$x_{44} = 9$$
$$x_{45} = 61$$
$$x_{46} = -11$$
$$x_{47} = 3$$
$$x_{48} = 7$$
$$x_{49} = -135$$
$$x_{50} = 25$$
$$x_{51} = -49$$
$$x_{52} = -45$$
$$x_{53} = 95$$
$$x_{54} = 5$$
$$x_{55} = 103$$
$$x_{56} = 75$$
$$x_{57} = 0$$
$$x_{58} = 73$$
$$x_{59} = 13$$
$$x_{60} = 667$$
$$x_{61} = -21$$
$$x_{62} = -81$$
$$x_{63} = -51$$
$$x_{64} = -13$$
$$x_{65} = -37$$
$$x_{66} = -17$$
$$x_{67} = -57$$
$$x_{68} = -43$$
$$x_{69} = 91$$
$$x_{70} = -7$$
$$x_{71} = -71$$
$$x_{72} = 215$$
so we need to solve the equation:
$$\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3$$
$$x_{2} = 0$$
$$x_{3} = 1$$
$$x_{4} = 3$$
Numerical solution
$$x_{1} = -29$$
$$x_{2} = 1$$
$$x_{3} = -33$$
$$x_{4} = -61$$
$$x_{5} = -59$$
$$x_{6} = -19$$
$$x_{7} = 71$$
$$x_{8} = -23$$
$$x_{9} = -1$$
$$x_{10} = -41$$
$$x_{11} = -2.99999999999964$$
$$x_{12} = 51$$
$$x_{13} = -2.99999999966621$$
$$x_{14} = -55$$
$$x_{15} = -35$$
$$x_{16} = -167$$
$$x_{17} = -31$$
$$x_{18} = -69$$
$$x_{19} = -25$$
$$x_{20} = -89$$
$$x_{21} = 399$$
$$x_{22} = -2.99999999998298$$
$$x_{23} = -27$$
$$x_{24} = 45$$
$$x_{25} = -67$$
$$x_{26} = -73$$
$$x_{27} = -47$$
$$x_{28} = -65$$
$$x_{29} = 59$$
$$x_{30} = -15$$
$$x_{31} = 69$$
$$x_{32} = -63$$
$$x_{33} = -75$$
$$x_{34} = -39$$
$$x_{35} = 93$$
$$x_{36} = -79$$
$$x_{37} = 85$$
$$x_{38} = -5$$
$$x_{39} = 57$$
$$x_{40} = 79$$
$$x_{41} = -53$$
$$x_{42} = -9$$
$$x_{43} = -2.999999999994$$
$$x_{44} = 9$$
$$x_{45} = 61$$
$$x_{46} = -11$$
$$x_{47} = 3$$
$$x_{48} = 7$$
$$x_{49} = -135$$
$$x_{50} = 25$$
$$x_{51} = -49$$
$$x_{52} = -45$$
$$x_{53} = 95$$
$$x_{54} = 5$$
$$x_{55} = 103$$
$$x_{56} = 75$$
$$x_{57} = 0$$
$$x_{58} = 73$$
$$x_{59} = 13$$
$$x_{60} = 667$$
$$x_{61} = -21$$
$$x_{62} = -81$$
$$x_{63} = -51$$
$$x_{64} = -13$$
$$x_{65} = -37$$
$$x_{66} = -17$$
$$x_{67} = -57$$
$$x_{68} = -43$$
$$x_{69} = 91$$
$$x_{70} = -7$$
$$x_{71} = -71$$
$$x_{72} = 215$$
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
$$\frac{\left(\frac{2 x}{3} + 1\right) \cos{\left(\frac{\pi x}{2} \right)}}{\left(x^{2} + 3 x\right)^{\frac{2}{3}}} - \frac{\pi \sqrt[3]{x^{2} + 3 x} \sin{\left(\frac{\pi x}{2} \right)}}{2} = 0$$
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
$$\frac{\left(\frac{2 x}{3} + 1\right) \cos{\left(\frac{\pi x}{2} \right)}}{\left(x^{2} + 3 x\right)^{\frac{2}{3}}} - \frac{\pi \sqrt[3]{x^{2} + 3 x} \sin{\left(\frac{\pi x}{2} \right)}}{2} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x^2 + 3*x)^(1/3)*cos((pi*x)/2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)}}{x}\right) = 0$$
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:
$$\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)} = \sqrt[3]{x^{2} - 3 x} \cos{\left(\frac{\pi x}{2} \right)}$$
- No
$$\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)} = - \sqrt[3]{x^{2} - 3 x} \cos{\left(\frac{\pi x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)} = \sqrt[3]{x^{2} - 3 x} \cos{\left(\frac{\pi x}{2} \right)}$$
- No
$$\sqrt[3]{x^{2} + 3 x} \cos{\left(\frac{\pi x}{2} \right)} = - \sqrt[3]{x^{2} - 3 x} \cos{\left(\frac{\pi x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd