Y=yx^3-3x+2 find the intervals where the function is concave up or down and inflection points

so hmmm since it's implicit differentiation, you'd get a function in x,y terms so. hmm if we check the graph of that second derivative, it seems to be asymptotic at x = -1, meaning an inflection point, from the denominator equaling 0

and it has an x-intercept at the origin, x = 0, another inflection point

now, the graph of the second derivative, from  x > 0, the value for "y" is positive all the way, with a horizontal asymptote at y  = 0
meaning the original function f(x) is concave UP  at x > 0

now, when -1 < x < 0, the value of "y" drops to the negatives, thus is concave DOWN  at -1 < x < 0

and if we pass the asymptotic inflection point of x = -1, to the left
x < -1, the value of "y" is on the positives, thus concave UP on that region

Idk idk idk idk idk idk idk idk