Question

Is `f(x)=4x^5-2x^4-9x^3-2x^2-6x` concave or convex at `x=-1`?

Answer 1

Concave (also called "concave down").

Explanation:

Concavity and convexity are determined by the sign of the second derivative:

• If `f''(-1)<0`, then the function is concave at `x=-1`.
• If `f''(-1)>0`, then the function is convex at `x=-1`.

Find the second derivative:

`f(x)=4x^5-2x^4-9x^3-2x^2-6x`
`f'(x)=20x^4-8x^3-27x^2-4x-6`
`f''(x)=80x^3-24x^2-54x-4`

Find the sign of the second derivative when `x=-1`:

`f''(-1)=80(-1)^3-24(-1)^2-54(-1)-4`

`=80(-1)-24(1)+54-4=-80-24+50=-54`

Since `f''(-1)<0`, the function is concave at `x=-1`. This means that it will resemble the `nn` shape. We can check a graph of `f(x)`:

graph{4x^5-2x^4-9x^3-2x^2-6x [-5, 5, -26.45, 19.8]}