Question

Is `f(x)=-2x^4-x^3+4x+4` concave or convex at `x=-1`?

Answer 1

concave

Explanation:

Find the second derivative and then compute `f''(-1)`. If the answer is positive, the graph is convex. If it's negative, the graph is concave at that point.

`f'(x)=-8x^3-3x^2+4`

`f''(x)=-24x^2-6x`

`f''(-1)=-24(-1)^2-6(-1)=-24+6=-18`

Since `-18<0`, the function is concave at `x=-1`.

Note: "concave" can also be called "concave downward" or "convex upward". It means the graph is heading downwards.