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

Aug 11, 2018

Convex $x \in \left(- 12 , 1\right) \mathmr{and} \left(1 , 3\right)$ and
concave $x \notin \left[- 12 , 3\right]$

#### Explanation:

Continuous and differentiable ( polynomial )

$f = {x}^{3} \left(1 + O \left(\frac{1}{x}\right)\right)$.

As $x \to \pm \infty , f \to \pm \infty ,$ respectively.

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

So, it is convex, for $x \in \left(- 12 , 1\right) \mathmr{and} \left(1 , 3\right)$. 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,

$f = \left(x - {x}_{1}\right) \left(x - {x}_{2}\right) \left(x - {x}_{3}\right) \left(x - {x}_{n}\right) , {x}_{i} < {x}_{i + 1}$

is convex,

$x \in \left({x}_{1} , {x}_{n}\right)$, sans $x = {x}_{i} , i = 1 , 2 , 3 , \ldots , n$ and concave, for

$x \notin \left[{x}_{1} , {x}_{n}\right]$