Signs diagram for second derivative

Question

For this problem I have to draw a sign diagram for the second derivative of the function x^(1/3). I have to determine whether the graph is concave up or down.

Here is my confusion, when I get the second derivative, which is -2/9x^(5/3), there is no solution when I set it equal to zero, which would've helped me get some intervals, and so I decided I was going to use the domain of the original function to create my signs diagram, but when I attempted to insert zero into the second derivative it didn't work, which made me realize that I might actually have to use the domain of the second derivative to create my signs diagram, and I'm kind of confused because that's not what I was doing before because I usually just used the domain of the original function to solve these problems. Can someone help explain.

Answer 1

when x>0, y"<0, so the curve is concave down

whenx<0, y">0 so the curve is concave up

at x=0, there's an inflection point (0,0), where the curve changes from concave up to concave down

y=x^(1/3) is the same as y^3 = x which is the inverse function of y=x^3

it's the curve reflected across the diagonal line y=x,

points (x,y) become (y,x)

in inflection point (0,0) remains (0,0)

you may be familiar with the cubic polynomial graph of y=x^3. Juar flip it around the 45 degree diagonal line through the origin and you get y=x^(1/3).