Nature of roots of a quartic

Main post link -> https://brilliant.org/mathematics-problem/matts-tangent/?group=sShEatfVqK07

I wonder if anyone can answer my question posted to the top-rated solution?

In case the link doesn't work, the given solution said:

"Let p(x)=x4−2x3−9x2+2x+8 and q(x)=ax+b. By sketching the graph of p, we can easily see that the only way that q can be tangent to p in two distinct points is if q is right at the bottom of p, touching each 2 down-humps in a different point, let us say, c and d.

Now let us sketch r(x)=p(x)−q(x). Because q is right at the bottom of p, the difference p(x)−q(x) is always positive, except for x=c and x=d, which make it go to zero. Therefore, r is a polynomial with only 2 real roots c and d. Furthermore, r is always positive when we get closer to c and to d, which means both have multiplicity 2 or greater."

and I am looking for explanation of the last sentence.

Easy Math Editor

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Comments

Note that the url that you get for the problem is customized to you. In order for others to view it, you have to use the url provided in "Share this problem". I have updated the link that you provided in this discussion.

Calvin Lin Staff - 7 years, 9 months ago

Thanks Calvin. I wasn't sure if it was or not.

Matt McNabb - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$