Find all local maximum points, local minimum points, and points of inflection on the curve y=-3x

^{4}-8x

^{3}+18x

^{2}
What I did: First derivative is y'=-12x

^{3}-24x

^{2}+36x

Then I set it equal to 0

Then I factored it out: -12x(x

^{2}+2x-3) = -12x(x-1)(x-2), so the critical numbers are x=0, x=1, and x=2

After getting the critical numbers, I did the second derivative

y''=36x

^{2}-58x+36

I used the second derivative test, and they end up all positive, so it's all min,

So I'm a bit confused.

Isn't this the way to find

**absolute minimum**,

**instead of local minimum** o.o

And how do I go on from there