Question

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

Answer 1

Concave (this is also called concave down).

Explanation:

The concavity or convexity of a function are determined by the sign of the second derivative.

• 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 of the function is a simple application of the power rule.

`f(x)=9x^3+2x^2-2x-2`

`f'(x)=27x^2+4x-2`

`f''(x)=54x+4`

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

`f''(-1)=-54+4=-50`

Since this is `<0`, the function is concave at `x=-1`. Concavity means that the function resembles the `nn` shape. We can check the graph of `f(x)`:

graph{9x^3+2x^2-2x-2 [-3, 3, -15, 15]}