Question

How do you determine whether the function `f(x) = -x^4-9x^3+2x+4` is concave up or concave down and its intervals?

Answer 1

`f(x)` is concave on `(-oo,-4.5)` and `(0,oo)`, and `f(x)` is convex on `(-4.5,0)`.

Explanation:

To find where a function is concave up, find where the second derivative of the function is positive.

`f(x)=-x^4-9x^3+2x+4`
Find `f'(x)`:
`f'(x)=-4x^3-27x^2+2`
Next, find `f''(x)`:
`f''(x)=-12x^2-54x`
`f''(x)=(-6x)(2x+9)`
Set `f''(x)` equal to zero to find inflection points
`0=(-6x)(2x+9)`
`x=0`, `x=-4.5`

After checking the signs of values around these numbers, we find that `f''(x)` is positive on `(-4.5,0)` i.e. convex and `f''(x)` is negative on `(-oo,-4.5)uu(0,oo)` i.e. concave.
graph{-x^4-9x^3+2x+4 [-10, 5, -1000, 1000]}