Calculus Concept problem

Consider the odd function f that is continuous,differentiable, and has the function values shown in the table.





(a) Determine f (4).

(b) Determine f (-3).

(c) Plot the points and make a possible sketch of the graph off on the interval [-6,6]. What is the smallest critical number of critical points in the interval? Explain.


(d) Does there exist at least one real number c in the interval (-6,6) where f '(c) = -1? Explain.


(e) Is it possible that f (x) does not exist? Explain.

(f) Is it necessary that f '(x) exists at x = 2? Explain.

I've completed a, b, and c, but I don't know how to explain the rest. Thanks for any help anyone can lend!

sbart said:

Consider the odd function f that is continuous,differentiable, and has the function values shown in the table.

x​	-5​	-4​	-1​	0​	2​	3​	6​
f(x)​	1​	3​	2​	0​	-1​	-4​	0​

(d) Does there exist at least one real number c in the interval (-6,6) where f '(c) = -1? Explain.
(e) Is it possible that

f (x) does not exist? Explain.

(f) Is it necessary that f '(x) exists at x = 2? Explain.

Click to expand...

d) Use the mean value theorem
\(\displaystyle \dfrac{f(2)-f(-1)}{2-(-1)}=~?\)


(e) Note that \(\displaystyle f\) is continuous. What does that have to do with limits?

You think about (f).