Is #f(x)=-4x^5-2x^4-5x^3+2x-31# concave or convex at #x=-2#?

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Answer 1 · 2015-12-03T02:23:37

convex

If #f''(-2)<0#, the function is concave at that point.
If #f''(-2)>0#, the function is convex at that point.

#f'(x)=-20x^4-8x^3-15x^2+2#

#f''(x)=-80x^3-24x^2-15x+2#

#f''(-2)=576#

#576>0#, so #f(x)# is convex at the point #(-2,101)#.

graph{-4x^5-2x^4-5x^3+2x-31 [-43, 61, -44.06, 7.94]}

Notice how the graph does appear to be convex (making a #uu# shape) at #x=-2#.