Question

Is `f(x) = -3x^4+x^3 - 4x^2+x-4` concave or convex at `x=11`?

Answer 1

The answer lies in the sign of the second derivative: the function is convex if the second derivative is positive, and concave otherwise. So, let's compute it.

`f'(x) = -12x^3+3x^2-8x+1`

`f''(x) = -36x^2+6x-8`

Now evaluate `f''(11)`:

`f''(11)=-36*(11^2) + 6*11 - 8 = -36*121 +66-8`, which is clearly negative (it's `-4298` to be precise), so the function is concave in that point.