Question

Sketch the graph of a function that satisfies the given conditions:
f(0)=0
f'(-2)=f'(1)=f'(9)=0
lim x->x f(x)=0
lim x->x f(x)=-infinity
f'(x)<0 on (-infinity,-2), (1,6), and (9,infinity)
f'(x)>0 on (-2,1) and (6,9)
f''(x)>0 on (-infinity,0) and (12, infinity)
f''(x)<0 on (0,6) and (6,12)

Answer 1

Time to get out pen and paper, or to open up the Draw utility (below), and start drawing in the info given. The graph of this function goes thru (0,0).
The first derivative of this function is zero at -2, 1 and 9; this means that the graph has a horiz. tangent line at each of these three x values.

Please share a preliminary graph, and then I'd be glad to continue with more suggestions.