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> Suppose that $f(x)$ is a polynomial with degree $4$. Prove that $f(x)$ has either $0$ or exactly $2$ inflection points.
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So I let $f(x) = ax^4 + bx^3 + cx^2 + d$ as is the convention, and found the $f''(x)$ which is $12ax^2 + 6bx + 2c$. I think $D = 9b^2 - 24ac$, and if $D>0$, there are 2 inflection points, if $D=0$ there is 1 inflection point and $D<0$ there is 0 inflection point, and I've also used the quadratic formula to get that $x = \frac{-3\pm\sqrt{9b^2 - 24ac}}{12a}$, but I don't know what to do next...