Question

Is `f(x) =(x-5)^3+12x^2+5x-13` concave or convex at `x=0`?

Answer 1

It's concave.

Explanation:

To find it out, you need to compute the second derivative: we have

`f(x) = (x-5)^3+12x^2+5x-13`

`f'(x) = 3(x-5)^2+24x+5`

`f''(x) = 6(x-5)+24 = 6x-30+24 = 6x-6 = 6(x-1)`

At `x=0`, you have `f''(0) = 6(0-1) = -6`

Since this value is negative, the function is concave.