Question

Is `f(x)=-x^5+3x^4-9x^3-2x^2-6x` concave or convex at `x=8`?

Answer 1

`f(x)` is concave at `x=8`. We know this by looking at the second derivative, which tells us about the concavity/shape of the graph.

Explanation:

When looking for the concavity of a function, it's best to find the second derivative, `f''(x)`, of the function, `f(x)`.

When `f''(x)<0`, the `f(x)` is concave
When `f''(x)>0`, the `f(x)` is convex

The first derivative of this function is:
`f'(x)=-5x^4+12x^3-27x^2-4x-6`

The second derivative is:
`f''(x)=-20x^3+36x^2-54x-4`

Plug in `x=8` to get:
`f''(8)=-20(8)^3+36(8)^2-54(8)-4`
`f''(8)=-8372`

Since `f''(8)<0`, the `f(x)` is concave at `x=8`.