I am working on a problem, but I got at some point. Please point out where I went wrong.
An owl is flying along a straight road, which runs north and south. The owl's position at time x hous is p(x) miles north of the start of the road, where p(x)=x^3 - 12x
a) Find all critical points.
So I first find the derivative of the function, then set to zero and solve for x.
p'(x) = 3x^2 - 12
3x^2 - 12 = 0
3x^2 = 12
x = +/-2
b) Determine all intervals of increase and decrease of p.
The intervals are (-infinity, -2) (-2,2) and (2, infinity)
I picked random numbers between the intervals to determine the increase and decrease of p.
f '(-3) = 15
f '(0) = -12
f '(5) = 38
So, between the intervals ((-infinity, -2) and (2, infinity, p is increasing.
On the interval (-2,2), p is decreasing.
c) Find all local maxima and minima of p.
f(2) = (2^3) - (12*2) = -16
f(-2) = (-2^3) - (12*-2) = 32
Local min: (2,-16)
Local max: (-2, 32)
d) Determine all intervals of concavity and inflection points of the graph of p.
I know I have to find the second derivative. So f"(x) = 9x
How do I continue?
e) Make a sketch of the graph of p.
Mine looks like a negative parabola, with its midpoint at (2,4).
Please tell me how I am doing on this problem, and be detailed.
PS. My work is in purple, so it is easier to read.