Question

Is `f(x)=-x^3-12x^2-14x-2` concave or convex at `x=0`?

Answer 1

Concave at `x = 0`

Explanation:

The test for convexity/concavity is to check the value of the 2nd derivative `f''(x)`

• If `f''(0) > 0` then convex at `x = 0`

• If `f''(0) < 0` then concave at `x = 0`

Find the 2nd derivative :

`f(x) = - x^3 -12x^2 - 14x -2`

`f'(x) = -3x^2 -24x - 14`

`f''(x) = 6x - 24`

Now `f''(0) = 0 - 24 = -24 < 0`

Hence `f(x)` is concave at `x = 0`.