Question

How do you determine whether the function `f(x) = 6x^3+54x-9` is concave up or concave down and its intervals?

Answer 1

You can study the sign of the second derivative:

Explanation:

You start with the first derivative:
So:
`f'(x)=18x^2+54`
then:
`f''(x)=36x`
set `f''(x)>0`
so:
`36x>0` when `x>0`
or visually:

Where `x=0, y=-9` will be an infection point.
Plotting the graph you can confirm this: