is this right?

[terry33199]

here is a graph of f '(x)
f is increasing on (3,6) and (6,8)
f is concave down on (0,1.5) and (4.5,6) and (6,9)
the critical numbers of f are x=0, x=3, x=6, x=8
the values of x that f has a local maximum is x=8
the values of x that f has a local minimum is x=3

 

[pka]

It all look correct.
Be sure that you check your textbook's definition for critical numbers
Some would include x=6 and others not.

 

[terry33199]

what would the graph of f(x) look like if f is continuous on [0,9] and f(0)=0

 

[Deleted member 4993]

Hint:

Section of the function, in the domain [0,6), would look like a cosine curve.