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) + 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.