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Plotting a function graph/ 3*sin(x)*cos(x)

Graphing y = 3*sin(x)*cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 3*sin(x)*cos(x)
f(x)=3sin⁡(x)cos⁡(x)f{\left(x \right)} = 3 \sin{\left(x \right)} \cos{\left(x \right)} 
f = (3*sin(x))*cos(x)
The graph of the function
02468-8-6-4-2-10105-5
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:
3sin⁡(x)cos⁡(x)=03 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Numerical solution
x1=−31.4159265358979x_{1} = -31.4159265358979
x2=51.8362787842316x_{2} = 51.8362787842316
x3=−36.1283155162826x_{3} = -36.1283155162826
x4=4.71238898038469x_{4} = 4.71238898038469
x5=89.5353906273091x_{5} = 89.5353906273091
x6=36.1283155162826x_{6} = 36.1283155162826
x7=−59.6902604182061x_{7} = -59.6902604182061
x8=−42.4115008234622x_{8} = -42.4115008234622
x9=21.9911485751286x_{9} = 21.9911485751286
x10=43.9822971502571x_{10} = 43.9822971502571
x11=15.707963267949x_{11} = 15.707963267949
x12=−53.4070751110265x_{12} = -53.4070751110265
x13=−80.1106126665397x_{13} = -80.1106126665397
x14=7.85398163397448x_{14} = 7.85398163397448
x15=−29.845130209103x_{15} = -29.845130209103
x16=−6.28318530717959x_{16} = -6.28318530717959
x17=78.5398163397448x_{17} = 78.5398163397448
x18=−10131.6363078271x_{18} = -10131.6363078271
x19=−23.5619449019235x_{19} = -23.5619449019235
x20=−17.2787595947439x_{20} = -17.2787595947439
x21=14.1371669411541x_{21} = 14.1371669411541
x22=58.1194640914112x_{22} = 58.1194640914112
x23=94.2477796076938x_{23} = 94.2477796076938
x24=12.5663706143592x_{24} = 12.5663706143592
x25=31.4159265358979x_{25} = 31.4159265358979
x26=−51.8362787842316x_{26} = -51.8362787842316
x27=−39.2699081698724x_{27} = -39.2699081698724
x28=80.1106126665397x_{28} = 80.1106126665397
x29=81.6814089933346x_{29} = 81.6814089933346
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=64.4026493985908x_{31} = 64.4026493985908
x32=590.619418874881x_{32} = 590.619418874881
x33=67.5442420521806x_{33} = 67.5442420521806
x34=−20.4203522483337x_{34} = -20.4203522483337
x35=0x_{35} = 0
x36=−58.1194640914112x_{36} = -58.1194640914112
x37=−50.2654824574367x_{37} = -50.2654824574367
x38=−81.6814089933346x_{38} = -81.6814089933346
x39=86.3937979737193x_{39} = 86.3937979737193
x40=−64.4026493985908x_{40} = -64.4026493985908
x41=−14.1371669411541x_{41} = -14.1371669411541
x42=−48.6946861306418x_{42} = -48.6946861306418
x43=26.7035375555132x_{43} = 26.7035375555132
x44=23.5619449019235x_{44} = 23.5619449019235
x45=−87.9645943005142x_{45} = -87.9645943005142
x46=6.28318530717959x_{46} = 6.28318530717959
x47=−61.261056745001x_{47} = -61.261056745001
x48=−119.380520836412x_{48} = -119.380520836412
x49=−37.6991118430775x_{49} = -37.6991118430775
x50=−7.85398163397448x_{50} = -7.85398163397448
x51=34.5575191894877x_{51} = 34.5575191894877
x52=45.553093477052x_{52} = 45.553093477052
x53=65.9734457253857x_{53} = 65.9734457253857
x54=20.4203522483337x_{54} = 20.4203522483337
x55=−28.2743338823081x_{55} = -28.2743338823081
x56=−67.5442420521806x_{56} = -67.5442420521806
x57=72.2566310325652x_{57} = 72.2566310325652
x58=−43.9822971502571x_{58} = -43.9822971502571
x59=73.8274273593601x_{59} = 73.8274273593601
x60=−9.42477796076938x_{60} = -9.42477796076938
x61=−75.398223686155x_{61} = -75.398223686155
x62=113.097335529233x_{62} = 113.097335529233
x63=−95.8185759344887x_{63} = -95.8185759344887
x64=−97.3893722612836x_{64} = -97.3893722612836
x65=87.9645943005142x_{65} = 87.9645943005142
x66=42.4115008234622x_{66} = 42.4115008234622
x67=95.8185759344887x_{67} = 95.8185759344887
x68=59.6902604182061x_{68} = 59.6902604182061
x69=29.845130209103x_{69} = 29.845130209103
x70=100.530964914873x_{70} = 100.530964914873
x71=−73.8274273593601x_{71} = -73.8274273593601
x72=−40.8407044966673x_{72} = -40.8407044966673
x73=48.6946861306418x_{73} = 48.6946861306418
x74=28.2743338823081x_{74} = 28.2743338823081
x75=70.6858347057703x_{75} = 70.6858347057703
x76=−94.2477796076938x_{76} = -94.2477796076938
x77=−89.5353906273091x_{77} = -89.5353906273091
x78=−21.9911485751286x_{78} = -21.9911485751286
x79=56.5486677646163x_{79} = 56.5486677646163
x80=−65.9734457253857x_{80} = -65.9734457253857
x81=−15.707963267949x_{81} = -15.707963267949
x82=−45.553093477052x_{82} = -45.553093477052
x83=−1.5707963267949x_{83} = -1.5707963267949
x84=−83.2522053201295x_{84} = -83.2522053201295
x85=−86.3937979737193x_{85} = -86.3937979737193
x86=92.6769832808989x_{86} = 92.6769832808989
x87=50.2654824574367x_{87} = 50.2654824574367
x88=37.6991118430775x_{88} = 37.6991118430775
x89=1.5707963267949x_{89} = 1.5707963267949
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (3*sin(x))*cos(x).
3sin⁡(0)cos⁡(0)3 \sin{\left(0 \right)} \cos{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−3sin⁡2(x)+3cos⁡2(x)=0- 3 \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\pi}{4}
The values of the extrema at the points:
 -pi        
(----, -3/2)
  4         

 pi      
(--, 3/2)
 4


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=−π4x_{1} = - \frac{\pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
[−π4,π4]\left[- \frac{\pi}{4}, \frac{\pi}{4}\right]
Increasing at intervals
(−∞,−π4]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\pi}{4}, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−12sin⁡(x)cos⁡(x)=0- 12 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}

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
[−π2,0]∪[π2,∞)\left[- \frac{\pi}{2}, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
(−∞,−π2]∪[0,π2]\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \frac{\pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(3sin⁡(x)cos⁡(x))=⟨−3,3⟩\lim_{x \to -\infty}\left(3 \sin{\left(x \right)} \cos{\left(x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡x→∞(3sin⁡(x)cos⁡(x))=⟨−3,3⟩\lim_{x \to \infty}\left(3 \sin{\left(x \right)} \cos{\left(x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (3*sin(x))*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3sin⁡(x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3sin⁡(x)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{3 \sin{\left(x \right)} \cos{\left(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:
3sin⁡(x)cos⁡(x)=−3sin⁡(x)cos⁡(x)3 \sin{\left(x \right)} \cos{\left(x \right)} = - 3 \sin{\left(x \right)} \cos{\left(x \right)}
- No
3sin⁡(x)cos⁡(x)=3sin⁡(x)cos⁡(x)3 \sin{\left(x \right)} \cos{\left(x \right)} = 3 \sin{\left(x \right)} \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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