Is #f(x) =-x^3-(x-2)(x+7)# concave or convex at #x=-3#?

1 Answer
Jim G.
Apr 6, 2016

convex at x = -3

Explanation:

To determine if a function is concave / convex at f(a) , require to find the value of f''(a).

• If f''(a) > 0 then f(x) is convex at x = a

• If f''(a) < 0 then f(x) is concave at x = a

rewrite f(x) by expanding the brackets

f(x) = #-x^3 - (x^2 +5x - 14) = -x^3 -x^2 - 5x + 14 #

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

and f''(x) = -6x - 2

#rArr f''(-3) = -6(-3) - 2 = 16 #

Since f''(-3) > 0 then f(x) is convex at x = -3

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