Is #f(x)=-2x^5-2x^4+5x-45# concave or convex at #x=-2#?

1 Answer
Nov 19, 2015

Answer:

A function (or its graph) can be said to be concave or convex on an interval. This function is convex near #-2#. (In some open interval containing #-2#.)

Explanation:

A necessary and sufficient condition for #f# to be convex on an interval is that #f''(x) >0# for all #x# in the interval.

In this case,

#f''(x) = -40x^3-24x^2#.

So, #f''(-2) = -40(-8)-24(4) <0#

#f''(x)# is continuous near #-2#, so #f''(x) <0# for #x# near #-2# and #f# is convex near #-2#.

(If you have been given a definition of #f# is "convex at a number, #a#", then I would guess you'll say #f# is convex at #-2#.)