For what values of x is #f(x)=(x+6)(x-1)(x+3)# concave or convex?

1 Answer
George C.
Dec 8, 2017

#f(x)# is concave for #x in (-oo, -8/3)# and convex for #x in (-8/3, oo)#

Explanation:

A function is convex where its second derivative is positive and concave where its second derivative is negative. If its second derivative is #0#, then the function may be concave, convex or neither. Commonly such will be a point of inflexion.

Given:

#f(x) = (x+6)(x-1)(x+3)#

#color(white)(f(x)) = x^3+8x^2+9x-18#

We find:

#f'(x) = 3x^2+16x+9#

and:

#f''(x) = 6x+16#

Hence #f(x)# is concave for #x in (-oo, -8/3)# and convex for #x in (-8/3, oo)#

It has a point of infexion at #x=-8/3#

graph{(x+6)(x-1)(x+3) [-8.21, 1.79, -25, 22]}

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