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

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Answer 1 · 2018-08-11T03:44:50

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

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 ]#