How do you determine where the given function #f(x) = (x+3)^(2/3) - 6# is concave up and where it is concave down?

1 Answer
May 1, 2015

In order to investigate concavity, we'll look at the sign of the second derivative.

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

#f'(x) = 2/3(x+3)^(-1/3)#

Notice that #x=-3# there is a cusp at which the tangent becomes vertical. (The derivative goes to #oo#)

#f''(x) = -2/9(x+3)^(-4/3) = (-2)/(9(x+3)^(4/3)) = (-2)/(9(root(3)(x+3)^4)#

The only place where #f''# might change sign is at #x=-3#.

But clearly the numerator is always negative, and the denominator, being a positive times a 4th power, is always positive.
So #f''# is always negative where it is defined.

The graph is concave down on #(-oo,-3)# and on #(-3,oo)#.

Because of the cusp at ##x=0#, we cannot combine these two intervals.

Here's the graph of #f#

graph{(x+3)^(2/3) - 6 [-18.74, 13.3, -15.11, 0.92]}