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Plotting a function graph/ cos(x)^4

Graphing y = cos(x)^4

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          4   
f(x) = cos (x)
f(x)=cos⁡4(x)f{\left(x \right)} = \cos^{4}{\left(x \right)} 
f = cos(x)^4
The graph of the function
02468-8-6-4-2-101002
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:
cos⁡4(x)=0\cos^{4}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=−23.5624641310095x_{1} = -23.5624641310095
x2=89.5358464044975x_{2} = 89.5358464044975
x3=64.4022227094897x_{3} = 64.4022227094897
x4=−67.5448065308884x_{4} = -67.5448065308884
x5=4.71186425026897x_{5} = 4.71186425026897
x6=80.1114831041243x_{6} = 80.1114831041243
x7=−45.5536354157268x_{7} = -45.5536354157268
x8=95.8191611950437x_{8} = 95.8191611950437
x9=86.393394845477x_{9} = 86.393394845477
x10=51.8368135303721x_{10} = 51.8368135303721
x11=20.4198789484825x_{11} = 20.4198789484825
x12=−14.1367106446029x_{12} = -14.1367106446029
x13=−58.1190619806665x_{13} = -58.1190619806665
x14=14.1376276021486x_{14} = 14.1376276021486
x15=−39.2699360040648x_{15} = -39.2699360040648
x16=67.5449867319022x_{16} = 67.5449867319022
x17=−80.110238235034x_{17} = -80.110238235034
x18=−1.57129267637417x_{18} = -1.57129267637417
x19=29.8456391715984x_{19} = 29.8456391715984
x20=29.8446819952113x_{20} = 29.8446819952113
x21=36.1288337410562x_{21} = 36.1288337410562
x22=−7.85359055632515x_{22} = -7.85359055632515
x23=−29.8448005950739x_{23} = -29.8448005950739
x24=−83.2518382112953x_{24} = -83.2518382112953
x25=−61.2608756650826x_{25} = -61.2608756650826
x26=42.4110507437587x_{26} = 42.4110507437587
x27=7.85446444955012x_{27} = 7.85446444955012
x28=26.7027138657113x_{28} = 26.7027138657113
x29=−17.279021473451x_{29} = -17.279021473451
x30=−89.5359774768786x_{30} = -89.5359774768786
x31=58.1200312868449x_{31} = 58.1200312868449
x32=−36.1278861189969x_{32} = -36.1278861189969
x33=−51.8359986082336x_{33} = -51.8359986082336
x34=−95.8183696553645x_{34} = -95.8183696553645
x35=−73.8271872272585x_{35} = -73.8271872272585
x36=73.8279875350762x_{36} = 73.8279875350762
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)^4.
cos⁡4(0)\cos^{4}{\left(0 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
−4sin⁡(x)cos⁡3(x)=0- 4 \sin{\left(x \right)} \cos^{3}{\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}
The values of the extrema at the points:
(0, 1)

 -pi     
(----, 0)
  2      

 pi    
(--, 0)
 2


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=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Maxima of the function at points:
x2=0x_{2} = 0
Decreasing at intervals
[−π2,0]∪[π2,∞)\left[- \frac{\pi}{2}, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−π2]∪[0,π2]\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \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
4(3sin⁡2(x)−cos⁡2(x))cos⁡2(x)=04 \left(3 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−5π6x_{1} = - \frac{5 \pi}{6}
x2=−π2x_{2} = - \frac{\pi}{2}
x3=−π6x_{3} = - \frac{\pi}{6}
x4=π6x_{4} = \frac{\pi}{6}
x5=π2x_{5} = \frac{\pi}{2}
x6=5π6x_{6} = \frac{5 \pi}{6}

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
[−5π6,−π6]∪[π6,∞)\left[- \frac{5 \pi}{6}, - \frac{\pi}{6}\right] \cup \left[\frac{\pi}{6}, \infty\right)
Convex at the intervals
(−∞,−5π6]\left(-\infty, - \frac{5 \pi}{6}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞cos⁡4(x)=⟨0,1⟩\lim_{x \to -\infty} \cos^{4}{\left(x \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→∞cos⁡4(x)=⟨0,1⟩\lim_{x \to \infty} \cos^{4}{\left(x \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 cos(x)^4, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡4(x)x)=0\lim_{x \to -\infty}\left(\frac{\cos^{4}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡4(x)x)=0\lim_{x \to \infty}\left(\frac{\cos^{4}{\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:
cos⁡4(x)=cos⁡4(x)\cos^{4}{\left(x \right)} = \cos^{4}{\left(x \right)}
- Yes
cos⁡4(x)=−cos⁡4(x)\cos^{4}{\left(x \right)} = - \cos^{4}{\left(x \right)}
- No
so, the function
is
even
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