Using First Derivative Test to find the solution for the problem

Question

Hi, I got a Calculus problem I just want to verify that I got right

1. Given f(x)=x^4 - 3x^3. Use the 1st Derivative Test to find the solution.
2. a) Find all x coords of critical pts and possible inflection pts.
3. b) Find all intervals over which f(x) is increasing, decreasing, concave up, and down

I got the critical points at x=0 and 9/4, possible inflection points at x=0 and x=3/2, and to find the intervals over which f(x) is increasing, decreasing, concave up or down, I just plug the inflection points on an x axis and find out whether it's +(con up) or -(con down) on each side of the inflect. pts, and to find the increasing, decreasing, I just do the same thing with critical pts, yes?

I put decreasing as (- infinity, 9/4) but apparently there was supposed to be a bracket around 9/4 (including), I'm not sure why?

And I have concave up in (- infity, 0) but concaving down in (0, infinity) is incorrect.. Can someone explain why?

Thank you

Answer 1

Graph the expression!

Yes, you have the critical points correctly, but 9/4 is a minimum and 0 is a point of inflection with a horizontal tangent, where the 2nd derivative is 0.