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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.

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    There is at least one, but there could be more than one.2012-09-07

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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.

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    Oh yes. I see it now. Thanks Jim Thanks @André.2012-09-07