TITLE: How to guess the number of inflection points?
QUESTION [8 upvotes]: I am asked to find fast the number of possible inflection points of:
$$y=(x-1)(x-2)^2(x-3)^4(x-4)^3$$
I know if the degree of any polynomial is even, its plot starts from the 2th quadrant to 1st quadrant of $\mathbb R^2$. This was what I could do fast. Any Ideas? Thanks.

REPLY [9 votes]: Just imagine what the graph looks like.  It starts above the $x$-axis, crosses below at $x=1$, is tangent to the $x$-axis at $x=2$ and $x=3$, and then crosses above at $x=4$, with an inflection point at $(4,0)$.  Thinking about the shape, I count:

One inflection point between $x=1$ and $x=2$,
Two inflection points between $x=2$ and $x=3$,
Two inflection points between $x=3$ and $x=4$, and
One inflection point at $x=4$.

Thus there are six inflection points.  This makes sense -- the second derivative should have eight zeroes, but two of them are at $x=3$, leaving six inflection points.