For what values of x is f(x)=(x+2)(x-3)(x+1) concave or convex?

1 Answer
Apr 13, 2017

The function is concave for x in (-oo,0)
The function is convex for x in (0,+oo)

Explanation:

We need

(uvw)'=u'vw+uv'w+uvw'

We must calculate the second derivative and determine the sign.

f(x)=(x+2)(x-3)(x+1)

f'(x)=(x-3)(x+1)+(x+2)(x+1)+(x+2)(x-3)

=x^2-2x-3+x^2+3x+2+x^2-x-6

=3x^2-7

f''(x)=6x

Therefore,

f''(x)=0 when x=0

This is the point of inflexion.

We can build a chart

color(white)(aaaa)Intervalcolor(white)(aaaaaa)(-oo,0)color(white)(aaaaaa)(0,+oo)

color(white)(aaaa)sign f''(x)color(white)(aaaaaaa)-color(white)(aaaaaaaaaaa)+

color(white)(aaaa) f(x)color(white)(aaaaaaaaaaaaa)nncolor(white)(aaaaaaaaaaa)uu