Is #f(x)=(x-3)(x-2)-x^2+x^3# concave or convex at #x=-1#?

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Answer 1 · 2017-10-01T01:19:40

Concave down

To determine if a function is concave up or down at a point, you need to examine the sign of the second derivative #f''(x)# at that point.

First, multiply out and simplify #f(x)#:

#f(x) = x^2-2x-3x+6-x^2+x^3 #
#f(x) = x^3 -5x + 6 #

Find #f'(x)# and then #f''(x)#:

#f'(x) = 3x^2 - 5#
#f''(x) = 6x#

At #x=-1#, #f''(-1) = -6 < 0#. Since the second derivative is negative at #x=-1#, that indicates that the method is concave down.

graph{(x-3)(x-2)-x^2+x^3 [-14.32, 14.15, -3.2, 11.04]}