Question

For what values of x is `f(x) = -sqrt(x^3-9x` concave or convex?

Answer 1

The function is concave on the interval `{-3, 0}`.

Explanation:

The answer is easily determined by viewing the graph:

graph{-sqrt(x^3 - 9x) [-4.8, 6.603, -4.618, 1.086]}

We already know that the answer is only real for the intervals `{-3,0}` and `{3, infty}`. Other values will result in an imaginary number, so they're out as far as finding concavity or convexity.

The interval `{3, infty}` does not change direction, so it can be neither concave nor convex. Thus the only possible answer is `{-3,0}`, which, as can be seen from the graph, is concave.