Question

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

Answer 1

Concave

Explanation:

While the function is undefined at `x = -1` there is nonetheless an answer as to convexity.

Concave vs. convex is a property of a function which is defined without respect to the point being analyzed, but rather with respect to the points on either side of it.

A function is convex with respect to a point on an interval if all of the possible line segments between points within that interval are above that point. Conversely, if all of the possible segments are below that point, it is convex. This technical defintion can be easily visualized by looking at a graph of a function. If it appears to form a "hill" then it is concave. If it appears to form a "trough" then it is convex.

Graphing your function, it is evident that it forms a hill, despite not being defined at `x = -1`. As such, it is a concave function.