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.
