Thread: Graph sketching, continuity, differentiation - Is this right?

(I would appreciate either an explanation as to how to solve the problems, or the correct answer so I could figure out what I did wrong.)

1. Let f(x) =

4-x^2 if x < 2
x^2 - 5x + 6 if 2 < x < 3
x if x > 3

a) Sketch the graph of f.

http://img293.imageshack.us/my.php?image=calcgraphgk1.png

b) At what numbers is f continuous?

- I'm not sure if I did this right, are you supposed to do something with limits to show what numbers are continuous?
(-∞, 2], [2, 3], [3, ∞)

c) At what numbers is f differentiable?

- Again, I've seen in some places where your answer has to include limits. Do you have to use limits in your work or something?
(-∞ ,∞)

d) Give a formula for f'.
f'(x) =

2x if x < 2
2x - 5 if 2 < x < 3
1 if x > 3

2. Let f(x) = x^3 - 9x^2 + 24x + 3
a) For what values of x does the graph of f have a horizontal tangent
- Okay I have no idea what to do for this one. I tried looking through my textbook and couldn't find anything. If someone could show me how to do this, that would be great.

b)
f''(x) = 9

2. Let f(x) = x^3 - 9x^2 + 24x + 3
a) For what values of x does the graph of f have a horizontal tangent
- Okay I have no idea what to do for this one. I tried looking through my textbook and couldn't find anything. If someone could show me how to do this, that would be great.

Since it is a cubic, it'll have 2 horizontal tangents.

Differentiate, set to 0 and solve for x.

The derivative will be a quadratic, which is easy to solve.

For that matter, just graph it.

So the rest of what I did is correct?

I apologize for double posting, but I really need to know whether or not I am correct for at least number 1.

Also, I would really like to know if I showed enough work, if not, how do I show more work?