At what points is the curve y=x^1/3*(x+3)^(2/3) concave up and concave down.?

The function we have is
y=x^(1/3)*(x+3)^(2/3).

This function is concave up when it
stops decreasing and starts to increase instead. It is concave down in the opposite
scenario when the function is no longer increasing but starts to
decrease.

y=x^(1/3)*(x+3)^(2/3)

y'
= [x^(1/3)]'*(x+3)^(2/3) + x^(1/3)*[(x+3)^(2/3)]'

y' =
(1/3)*x^(-2/3)*(x+3)^(2/3) + x^(1/3)*(2/3)*(x+3)^(-1/3)

y'
= (1/3)*[(x+3)/x]^(2/3) + (2/3)*[x/(x + 3)]^(1/3)

y' =
[(1/3)*(x+3) + (2/3)*x]/x^(2/3)*(x + 3)^(1/3)

y' = (x +
1)/x^(2/3)*(x + 3)^(1/3)

We see that there is only one
point of inflection which is where the function stops increasing or
decreasing.

The point of inflection is at x =
-1.

Here, the function is increasing on the right till x =
0, where the slope becomes infinity and decreasing on the left from x =
-3.

The curve is concave up in the region (-3
, 0). It is not concave down anywhere.