Question

Is `f(x)=-3x^3-x^2-3x+2` concave or convex at `x=-1`?

Answer 1

Convex.

Explanation:

The sign of the second derivative is indicative of the function's convexity or concavity:

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

Finding the second derivative requires a simple application of the power rule twice over:

`f(x)=-3x^3-x^2-3x+2`
`f'(x)=-27x^2-2x-3`
`f''(x)=-54x-2`

Find `f''(-1)`.

`f''(-1)=-54(-1)-2=54-2=52`

Since `f''(-1)>0`, `f(x)` is convex at `x=-1`. What this means is that the graph will form a `uu`-like shape. We can consult a graph of `f(x)`:

graph{-3x^3-x^2-3x+2 [-2, 2, -10, 15]}