Question

For what values of x is `f(x)=(x-1)(x-3)(x+12)` concave or convex?

Answer 1

Convex `x in ( - 12, 1 ) and ( 1, 3 )` and
concave `x notin [ - 12, 3 ]`

Explanation:

Continuous and differentiable ( polynomial )

`f = x^3 ( 1 +O ( 1/x ) )`.

As `x to +- oo, f to +- oo,` respectively.

Zeros of f are #x= - 12, 1, 3.

So, it is convex, for `x in ( - 12, 1 ) and( 1, 3 )`. See graph.
graph{y-(x+12)(x-1)(x-2)=0[-20 20 -1000 1000]}

Zoomed convex graph, for x in ( 1, 2 ):

graph{y-(x+12)(x-1)(x-2)=0[1 2 -10 10]}

In between successive leaving and returning to x-axis, the graph is

convex. For that matter,

`f = (x - x_1 ) ( x - x_2 )( x - x_3 )(x - x_n), x_i < x_(i+1)`

is convex,

`x in ( x_1, x_n)`, sans `x = x_i, i = 1, 2, 3, ..., n` and concave, for

`x notin [ x_1, x_n ]`