second derivative ques - help! (1 Viewer)

[lilkiwifruit]

1. By considering the sign of f'(x) and f''(x) sketch the shape of the curve y=f(x) in the given domains:

y= x^3 - 3x^2 + 6x - 2 from x=2 to x = 3

2. sketch the curve y=f(x) if:
f(0) = 3 , f(2) =7 and if in the domain x = 0 to x = 2, f'(x) > 0 and f"(x) <0


I don't know how to do these and the textbook isn't helping either as there are no sketches provided in the answers!
I would really appreciate your help

 

[lilkiwifruit]

I feel like an idiot because you guys have even thought of using rollercoasters to explain it and i still don't understand. Does anyone have a sketch? I'm soo confused now

 

[Riviet]

When f'(x) > 0, this means the gradient is positive and the curve is increasing from left to right like this:

a/
/

when f'(x) <0, the gradient is negative and the curve is decreasing from left to right like this this:

\
a\

When f "(x) >0, the curve is concave up with a shape like this:

\_/

When f "(x) <0, the curve is concave down with a shape like this:
a_
/ a\

P.S Ignore the a's, they are there to align my mini drawings.