Calculus

f'(x)= 0 (the gradient of f(x)) when x=o .
But x=0 is the y axis which is perpendicular to all lines of zero gradient.

x=0 is just the Y axis.

When you write x to the third power you should use a duncecap symbol i.e. the item over the 6 on your keyboard, and one assumes that's what you meant....

When you multiply (a*x^3 -b) and (c*x^3 - d) together you get a polynomial with x to the sixth and third power plus a constant so that your function looks like

f(x) = A*x^6 + B*x^3 + d

and when you take a first derivative of that the constant term goes out so that the thing evaluates to zero at the origin i.e. a tangent line would be the X axis and, of course, the Y axis is normal to that.

I don't think f(x) goes through the origin.
The intercept with the y axis is bd.

I'd figure given the context that a normal line would simply be one perpenducular to the tangent at the given point, so it shouldn't matter i.e. the tangent line and the y axis just cross eachother.