Hello,

I am working on a problem, but I got at some point. Please point out where I went wrong.

Problem:

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, wherep(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. Sof"(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.