Question

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

Answer 1

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]}