1. ## Shapes of Graph problem. please review my work!

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, 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.

2. ok

f"(x) = 9x

9x=0

x=0

take two values of x one more than 0 and the other less than 0
f"(1)=9 positive then f is concave up at [0,infinity)
f"(-1)= -9 negative then f is concave down at (- infinity , 0]

and I attack the graph as you can see the function is increasing from infinity to -2 and from -2 to 2 decreasing and more than 2 increasing and the point (0,0) you see that the curve change the concavity

so if you want to sketch you should first put the critical points and the inflection points then see the interval of increasing and decreasing and the concavity I write a lot of words

Attachment 11396