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Plotting a function graph/ sin^2(x)cos^2(x)

Graphing y = sin^2(x)cos^2(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          2       2   
f(x) = sin (x)*cos (x)
f(x)=sin⁡2(x)cos⁡2(x)f{\left(x \right)} = \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} 
f = sin(x)^2*cos(x)^2
The graph of the function
02468-8-6-4-2-10100.000.50
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:
sin⁡2(x)cos⁡2(x)=0\sin^{2}{\left(x \right)} \cos^{2}{\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=−20.4203521774723x_{1} = -20.4203521774723
x2=87.9645943351391x_{2} = 87.9645943351391
x3=−42.4115007432387x_{3} = -42.4115007432387
x4=15.7079633917898x_{4} = 15.7079633917898
x5=1.57079638652515x_{5} = 1.57079638652515
x6=12.5663704969137x_{6} = 12.5663704969137
x7=−94.247779486083x_{7} = -94.247779486083
x8=86.3937978937855x_{8} = 86.3937978937855
x9=−58.1194645366003x_{9} = -58.1194645366003
x10=−59.6902604569585x_{10} = -59.6902604569585
x11=−97.3893725907902x_{11} = -97.3893725907902
x12=6.28318528443138x_{12} = 6.28318528443138
x13=64.4026493150839x_{13} = 64.4026493150839
x14=−1.57079642013166x_{14} = -1.57079642013166
x15=56.5486676469942x_{15} = 56.5486676469942
x16=45.5530935075531x_{16} = 45.5530935075531
x17=34.5575190717885x_{17} = 34.5575190717885
x18=−51.8362786915081x_{18} = -51.8362786915081
x19=94.247779609353x_{19} = 94.247779609353
x20=36.1283154718409x_{20} = 36.1283154718409
x21=−80.1106125854791x_{21} = -80.1106125854791
x22=−86.3937978789102x_{22} = -86.3937978789102
x23=7.85398173011892x_{23} = 7.85398173011892
x24=−95.818575868455x_{24} = -95.818575868455
x25=78.5398162225044x_{25} = 78.5398162225044
x26=−15.7079632962205x_{26} = -15.7079632962205
x27=23.5619449483644x_{27} = 23.5619449483644
x28=20.4203521581227x_{28} = 20.4203521581227
x29=21.9911485851564x_{29} = 21.9911485851564
x30=−17.2787595621355x_{30} = -17.2787595621355
x31=−43.9822971747455x_{31} = -43.9822971747455
x32=−50.2654823342013x_{32} = -50.2654823342013
x33=−73.8274272808521x_{33} = -73.8274272808521
x34=67.5442420634706x_{34} = 67.5442420634706
x35=65.9734457525462x_{35} = 65.9734457525462
x36=−14.1371668484631x_{36} = -14.1371668484631
x37=−45.5530935761698x_{37} = -45.5530935761698
x38=73.8274274646672x_{38} = 73.8274274646672
x39=−37.6991118766796x_{39} = -37.6991118766796
x40=−75.3982237985682x_{40} = -75.3982237985682
x41=50.2654824463816x_{41} = 50.2654824463816
x42=42.4115007365289x_{42} = 42.4115007365289
x43=−89.5353907315491x_{43} = -89.5353907315491
x44=−39.2699081045218x_{44} = -39.2699081045218
x45=−28.2743337586152x_{45} = -28.2743337586152
x46=−9.42477807759933x_{46} = -9.42477807759933
x47=14.1371670778185x_{47} = 14.1371670778185
x48=100.530964798296x_{48} = 100.530964798296
x49=0x_{49} = 0
x50=−58.1194640062544x_{50} = -58.1194640062544
x51=−1.57079626356835x_{51} = -1.57079626356835
x52=−29.8451301000724x_{52} = -29.8451301000724
x53=4.71238898608896x_{53} = 4.71238898608896
x54=−87.9645943594276x_{54} = -87.9645943594276
x55=−83.2522051669813x_{55} = -83.2522051669813
x56=−53.4070752253874x_{56} = -53.4070752253874
x57=48.6946860958663x_{57} = 48.6946860958663
x58=29.8451303084991x_{58} = 29.8451303084991
x59=80.1106131511482x_{59} = 80.1106131511482
x60=−31.4159266517141x_{60} = -31.4159266517141
x61=51.8362788866811x_{61} = 51.8362788866811
x62=−100.530965206253x_{62} = -100.530965206253
x63=−97.3893723711949x_{63} = -97.3893723711949
x64=−21.9911485864927x_{64} = -21.9911485864927
x65=−65.9734457653935x_{65} = -65.9734457653935
x66=95.8185756842062x_{66} = 95.8185756842062
x67=−7.85398150696156x_{67} = -7.85398150696156
x68=−6.28318518328035x_{68} = -6.28318518328035
x69=95.8185760424586x_{69} = 95.8185760424586
x70=26.7035375390573x_{70} = 26.7035375390573
x71=92.6769832182628x_{71} = 92.6769832182628
x72=−23.5619449982306x_{72} = -23.5619449982306
x73=37.6991119665793x_{73} = 37.6991119665793
x74=28.2743338652921x_{74} = 28.2743338652921
x75=72.256630710694x_{75} = 72.256630710694
x76=−81.6814090370675x_{76} = -81.6814090370675
x77=59.690260541069x_{77} = 59.690260541069
x78=−36.128315427252x_{78} = -36.128315427252
x79=72.2566310277248x_{79} = 72.2566310277248
x80=−61.2610566398387x_{80} = -61.2610566398387
x81=70.6858338406532x_{81} = 70.6858338406532
x82=70.6858346557926x_{82} = 70.6858346557926
x83=89.5353906153414x_{83} = 89.5353906153414
x84=−64.402649310466x_{84} = -64.402649310466
x85=−72.2566309100272x_{85} = -72.2566309100272
x86=−67.5442421539445x_{86} = -67.5442421539445
x87=81.6814091152362x_{87} = 81.6814091152362
x88=43.9822971692691x_{88} = 43.9822971692691
x89=117.809724442492x_{89} = 117.809724442492
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)^2*cos(x)^2.
sin⁡2(0)cos⁡2(0)\sin^{2}{\left(0 \right)} \cos^{2}{\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
−2sin⁡3(x)cos⁡(x)+2sin⁡(x)cos⁡3(x)=0- 2 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 2 \sin{\left(x \right)} \cos^{3}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−3π4x_{2} = - \frac{3 \pi}{4}
x3=−π2x_{3} = - \frac{\pi}{2}
x4=−π4x_{4} = - \frac{\pi}{4}
x5=π4x_{5} = \frac{\pi}{4}
x6=π2x_{6} = \frac{\pi}{2}
x7=3π4x_{7} = \frac{3 \pi}{4}
The values of the extrema at the points:
(0, 0)

 -3*pi      
(-----, 1/4)
   4        

 -pi     
(----, 0)
  2      

 -pi       
(----, 1/4)
  4        

 pi      
(--, 1/4)
 4       

 pi    
(--, 0)
 2     

 3*pi      
(----, 1/4)
  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=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Maxima of the function at points:
x3=−3π4x_{3} = - \frac{3 \pi}{4}
x3=−π4x_{3} = - \frac{\pi}{4}
x3=π4x_{3} = \frac{\pi}{4}
x3=3π4x_{3} = \frac{3 \pi}{4}
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−π2]\left(-\infty, - \frac{\pi}{2}\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
2((sin⁡2(x)−cos⁡2(x))sin⁡2(x)−(sin⁡2(x)−cos⁡2(x))cos⁡2(x)−4sin⁡2(x)cos⁡2(x))=02 \left(\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} - 4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−2atan⁡(−1+22−2+2)x_{1} = - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x2=2atan⁡(−1+22−2+2)x_{2} = 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}
x3=−2atan⁡(1+2+22+2)x_{3} = - 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x4=2atan⁡(1+2+22+2)x_{4} = 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}
x5=−2atan⁡(−2+1+22−2)x_{5} = - 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x6=2atan⁡(−2+1+22−2)x_{6} = 2 \operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{2 - \sqrt{2}} \right)}
x7=−2atan⁡(−22+2+1+2)x_{7} = - 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}
x8=2atan⁡(−22+2+1+2)x_{8} = 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2} \right)}

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
[2atan⁡(1+2+22+2),∞)\left[2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2} \right)}, \infty\right)
Convex at the intervals
(−∞,−2atan⁡(−1+22−2+2)]\left(-\infty, - 2 \operatorname{atan}{\left(-1 + \sqrt{2} \sqrt{2 - \sqrt{2}} + \sqrt{2} \right)}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡2(x)cos⁡2(x))=⟨0,1⟩\lim_{x \to -\infty}\left(\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞(sin⁡2(x)cos⁡2(x))=⟨0,1⟩\lim_{x \to \infty}\left(\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)^2*cos(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡2(x)cos⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡2(x)cos⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)} \cos^{2}{\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:
sin⁡2(x)cos⁡2(x)=sin⁡2(x)cos⁡2(x)\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}
- Yes
sin⁡2(x)cos⁡2(x)=−sin⁡2(x)cos⁡2(x)\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}
- No
so, the function
is
even
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