Question

Is `f(x) =2x^3+(x+2)(x-4)` concave or convex at `x=-8`?

Answer 1

`f(x)` is convex at `x=-8`

Explanation:

Expanding `f(x)=2x^3+(x+2)(x-4)`
we get
`color(white)("XXX")f(x)=2x^3+x^2-2x-8`
from which (using the exponent rule for derivatives
`color(white)("XXX")f'(x)=6x^2+2x-2`
and
`color(white)("XXX")f''(x)=12x+2`

at `x=-8`
`color(white)("XXX")f''(-8)=12 * (-8) +2 < 0`
so `f(x)` is concave downward (convex) at `x=-8`

This means that although `f(x)` is still increasing at `x=-9`
the rate of increase is slowing down, so the graph is starting to curve towards what would become downward.
graph{x^3+(x+2)(x-4) [-28.4, 17.2, -19.88, 2.92]}