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critical and inflection points??

what are the critical points, inflection points, intervals of increase/decrease and concavity in the following?
i used data from monthly weather averages to obtain the equation
69.74+19.33sin(o.4688x-1.7050)
what are the critical points, inflection points, intervals of increase/decrease and concavity?
also the absolute extremas!

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2 Answers

f(x) = 69.74 + 19.33sin (0.4688x - 1.7050)
 
f'(x) = (0.4688)(19.33)cos (0.4688x - 1.7050) = 9.061904cos (0.4688x - 1.7050)
 
cos (0.4688x - 1.7050) = 0
 
0.4688x - 1.7050 = pi/2
 
0.4688x = pi/2 + 1.7050
 
x = (pi/2 + 1.7050)/.4688 = 6.9876
 
0.4688x - 1.7050 = 3pi/2
 
0.4688x = 3pi/2 + 1.7050
 
x = (3pi/2 + 1.7050)/.4688 = 13.689
 
Before x = 6.9876, the derivative is positive. When x is between 6.9876 & 13.689, the derivative is negative. When x is greater 13.689 the derivative is positive. Since the derivative is a cosine function, there are several relative minima and maxima because there are several points where the cosine function equals zero. With these two points, there is a maximum at 6.9876 & a minimum at 13.689
 
f"(x) = -4.2482sin (.4688x - 1.7050)
 
There zeroes for the second derivative are when:
 
.4688x - 1.7050 = 0, pi, 2p, 3pi...........kpi where k = whole number
 
x = (1.7050 + k*pi)/.4688 where k = 1,2,3,4............infinity
 
For every value of k, there is a point of inflection.
 
            0 < (0.4688x  -  1.7050)<(2*22/7)/.4688
 
              1.7050/.4688  <  x   < ( 1.7050/.4688) + (2*22/7)/.4688)
 
Period of sin wave = 13.4 radians or 767 degrees
Phase shift. --   208 degrees to the right.
 
Critical points --- x= 16 degrees, x=400 degrees , 784 degrees and so on
Points of inflection ---  x= 208 degrees ,  x= 592 degrees. and so on
Intervals of increase --  x= 16 degrees to 400 degrees
Intervals of decrease --  x= 400 degrees to 784 degrees
Concavity up --  x = 16 degrees to 208 degrees
Concavity down --  x = 208 degrees to 592 degrees
No absolute maxima,minima.
 
Surendra

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