Question

Is `f(x) = (x+3)^(2/3) - 6` concave or convex?

Answer 1

`f(x)` has a convex hypograph.

Explanation:

For `x in (-3,oo)` as we can see, `f(x) = (x+3)^(2/3) - 6` has a convex hypograph. This can be stated observing that in the same range, `f'(x) =2/(3 (3 + x)^(1/3))` has a positive strictly decreasing value. Further rigorous proof can be established by stating that if
`(x_1,f(x_1))` and `(x_2,f(x_2))` are two hypograph points, then

`(x_1+lambda(x_2-x_1),f(x_1+lambda(x_2-x_1)))` for `lambda in [0,1]` is also a hypograph point.

Attached the hypograph set