For what values of x is #f(x)=-x^3-4x^2+2x+5# concave or convex?

1 Answer
Jan 8, 2016

The curve is convex for #x > -4/3# and concave for #x<-4/3#

Explanation:

Use the second differential to decide concavity. If #f''(x) > 0# then the curve is convex, and if #f''(x) < 0 # then it is concave. Convex means the gradient is decreasing (going from positive to negative), and concave means the gradient is increasing (going form negative to positive).
#f'(x) = -3x^2 -8x +2#
#f''(x) = -6x -8#
#-6x - 8 > 0 iff x > 8/-6 iff -4/3#
The curve is convex for #x > -4/3# and concave for #x<-4/3#